Check to see if each first-order differential equation is linear, separable, both, or neither:a)&nbsp;&nbsp;dy&nbsp;&nbsp;dx&nbsp;+exy=x2y2b)&nbsp;y+sin⁡x=x3y′c)&nbsp;ln⁡x−x2y=xy′d)&nbsp;&nbsp;dy&nbsp;&nbsp;dx&nbsp;+cos⁡y=tan⁡x

Step 1A first-order differential equation is said to be separable if it can be written asdydx=f(x)g(y), f and g are known functions.A first-order differntial equation is said to be separable if it can be written asdydx+p(x)y=q(x), p(x) and q(x) are continuouse functions.a) Consider the differential equation dydx+exy=x2y2Rewrite the given differential equation as follows:dydx=x2y2−exy=y(x2y−ex)As the variables x and y cannot be separated and written as separate functions of x and y, given equation is not separable.Also, in the equation dydx+exy=x2y2, p(x)=ex but right hand side is not a function of x alone. Hence, it is not a linear equation.Therefore, the given differential equation is neither a separable equation nor a linear equation.b) Consider the differential equation y+sin⁡x=x3y′Rewrite the given differential equation as follows:y+sin⁡x=x3y′y+sin⁡x=x3dydxyx3+sin⁡xx3=dydxsin⁡xx3−dydx−yx3Thus, the given equation can be written as dydx+(−1x3)y=sin⁡xx3 wherep(x)=−1x3 and q(x)=sin⁡xx3Hence, it is a linear equation.Also, equation can be written as dy

Answer:(a)equation neither linear nor separable(b)equation is linear(c)equation is separable(d)equation is lineara) dydx+exy=x2y2To determine if this equation is linear or separable, we need to rewrite it in a standard form. Rearranging the terms, we get:dydx=x2y2−exyThis equation is neither linear nor separable since it cannot be expressed in the form dydx=f(x)g(y) or dydx=ax+by+c.b) y+sinx=x3y′Rearranging the terms, we have:x3y′−y=−sinxThis equation is linear since it can be expressed in the form y′+P(x)y=Q(x). Here, P(x)=−1x3 and Q(x)=−sinx.c) lnx−x2y=xy′Rearranging the terms, we obtain:xy′−x2y=lnxThis equation is separable since we can rewrite it in the form dydx=lnxx+xy.d) dydx+cosy=tanxThis equation is linear since it can be expressed in the form dydx+P(x)y=Q(x). Here, P(x)=cosy and Q(x)=tanx.

Step 1A first-order differential equation is said to be separable if it can be written asdydx=f(x)g(y), f and g are known functions.A first-order differntial equation is said to be separable if it can be written asdydx+p(x)y=q(x), p(x) and q(x) are continuouse functions.a) Consider the differential equation dydx+exy=x2y2Rewrite the given differential equation as follows:dydx=x2y2−exy=y(x2y−ex)As the variables x and y cannot be separated and written as separate functions of x and y, given equation is not separable.Also, in the equation dydx+exy=x2y2, p(x)=ex but right hand side is not a function of x alone. Hence, it is not a linear equation.Therefore, the given differential equation is neither a separable equation nor a linear equation.b) Consider the differential equation y+sin⁡x=x3y′Rewrite the given differential equation as follows:y+sin⁡x=x3y′y+sin⁡x=x3dydxyx3+sin⁡xx3=dydxsin⁡xx3−dydx−yx3Thus, the given equation can be written as dydx+(−1x3)y=sin⁡xx3 wherep(x)=−1x3 and q(x)=sin⁡xx3Hence, it is a linear equation.Also, equation can be written as dy

Answer:(a)equation neither linear nor separable(b)equation is linear(c)equation is separable(d)equation is lineara) dydx+exy=x2y2To determine if this equation is linear or separable, we need to rewrite it in a standard form. Rearranging the terms, we get:dydx=x2y2−exyThis equation is neither linear nor separable since it cannot be expressed in the form dydx=f(x)g(y) or dydx=ax+by+c.b) y+sinx=x3y′Rearranging the terms, we have:x3y′−y=−sinxThis equation is linear since it can be expressed in the form y′+P(x)y=Q(x). Here, P(x)=−1x3 and Q(x)=−sinx.c) lnx−x2y=xy′Rearranging the terms, we obtain:xy′−x2y=lnxThis equation is separable since we can rewrite it in the form dydx=lnxx+xy.d) dydx+cosy=tanxThis equation is linear since it can be expressed in the form dydx+P(x)y=Q(x). Here, P(x)=cosy and Q(x)=tanx.

We have an answer to "Check to see if each first-order differential equation..."

College
Algebra
Linear algebra
Forms of linear equations

pedzenekO

Answered question

2021-09-20

Check to see if each first-order differential equation is linear, separable, both, or neither:
a) $\frac{d y}{d x} + e^{x} y = x^{2} y^{2}$
b) $y + \sin x = x^{3} y^{'}$
c) $\ln x - x^{2} y = x y^{'}$
d) $\frac{d y}{d x} + \cos y = \tan x$

Answer & Explanation

Talisha

Skilled2021-09-21Added 93 answers

Step 1
A first-order differential equation is said to be separable if it can be written as

\frac{d y}{d x} = f (x) g (y), f

and

g

are known functions.
A first-order differntial equation is said to be separable if it can be written as

\frac{d y}{d x} + p (x) y = q (x), p (x)

and

q (x)

are continuouse functions.
a) Consider the differential equation

\frac{d y}{d x} + e^{x} y = x^{2} y^{2}

Rewrite the given differential equation as follows:

\frac{d y}{d x} = x^{2} y^{2} - e^{x} y

= y (x^{2} y - e^{x})

As the variables

x

and

y

cannot be separated and written as separate functions of

x

and

y

, given equation is not separable.
Also, in the equation

\frac{d y}{d x} + e^{x} y = x^{2} y^{2}, p (x) = e^{x}

but right hand side is not a function of x alone. Hence, it is not a linear equation.
Therefore, the given differential equation is neither a separable equation nor a linear equation.
b) Consider the differential equation

y + \sin x = x^{3} y^{'}

Rewrite the given differential equation as follows:

y + \sin x = x^{3} y^{'}

y + \sin x = x^{3} \frac{d y}{d x}

\frac{y}{x^{3}} + \frac{\sin x}{x^{3}} = \frac{d y}{d x}

\frac{\sin x}{x^{3}} - \frac{d y}{d x} - \frac{y}{x^{3}}

Thus, the given equation can be written as

\frac{d y}{d x} + (\frac{- 1}{x^{3}}) y = \frac{\sin x}{x^{3}}

where

p (x) = \frac{- 1}{x^{3}}

and

q (x) = \frac{\sin x}{x^{3}}

Hence, it is a linear equation.
Also, equation can be written as

\frac{d y}{This is helpful 30 Flagalenahelenash Expert 2023-05-13 Added 556 answers Step 1:a) \frac{d y}{d x} + e^{x y} = x^{2} y^{2} To determine if the equation is linear or separable, we need to rewrite it in a standard form. In this case, we observe that the equation is neither linear nor separable, as the terms involving y and y^{'} cannot be separated. Step 2: b) y + \sin x = x^{3} y^{'} To check linearity, we need to express the equation in the form y^{'} + P (x) y = Q (x) . We can rewrite the given equation as: y^{'} - \frac{1}{x^{3}} y = - \frac{\sin x}{x^{3}} The equation is linear, as it can be expressed in the desired form. Step 3: c) \ln x - x^{2} y = x y^{'} To determine if the equation is linear or separable, we need to rewrite it in a standard form. Rearranging the terms, we have: x y^{'} - x^{2} y = \ln x The equation is linear since it can be written in the form y^{'} + P (x) y = Q (x) . Step 4: d) \frac{d y}{d x} + \cos y = \tan x To check linearity, we need to express the equation in the form y^{'} + P (x) y = Q (x) . We can rewrite the given equation as: \frac{d y}{d x} + \cos y - \tan x = 0 This is helpful 0 Flagxleb123 Skilled 2023-05-13 Added 181 answers Answer: (a)equation neither linear nor separable (b)equation is linear (c)equation is separable (d)equation is linear a) \frac{d y}{d x} + e^{x y} = x^{2} y^{2} To determine if this equation is linear or separable, we need to rewrite it in a standard form. Rearranging the terms, we get: \frac{d y}{d x} = x^{2} y^{2} - e^{x y} This equation is neither linear nor separable since it cannot be expressed in the form \frac{d y}{d x} = f (x) g (y) or \frac{d y}{d x} = a x + b y + c . b) y + \sin x = x^{3} y^{'} Rearranging the terms, we have: x^{3} y^{'} - y = - \sin x This equation is linear since it can be expressed in the form y^{'} + P (x) y = Q (x) . Here, P (x) = - \frac{1}{x^{3}} and Q (x) = - \sin x . c) \ln x - x^{2} y = x y^{'} Rearranging the terms, we obtain: x y^{'} - x^{2} y = \ln x This equation is separable since we can rewrite it in the form \frac{d y}{d x} = \frac{\ln x}{x} + \frac{x}{y} . d) \frac{d y}{d x} + \cos y = \tan x This equation is linear since it can be expressed in the form \frac{d y}{d x} + P (x) y = Q (x) . Here, P (x) = \cos y and Q (x) = \tan x . This is helpful 0 Flagstar233 Skilled 2023-05-13 Added 403 answers a) \frac{d y}{d x} + e^{x y} = x^{2} y^{2} To determine if this equation is linear or separable, we need to check its form. The equation is neither linear nor separable because the term e^{x y} appears. Therefore, the equation is neither linear nor separable. b) y + \sin x = x^{3} y^{'} Rearranging the equation, we have x^{3} y^{'} - y = - \sin x . To check if it's linear or separable, we need to see if we can isolate y^{'} or y on one side. Since y^{'} appears with x^{3}, the equation is not separable. However, we can rewrite it as: y^{'} - \frac{y}{x^{3}} = - \frac{\sin x}{x^{3}} Now we can see that the equation is linear. Applying an integrating factor method, the solution is: y (x) = \frac{\int x^{- 3} e^{\int x^{- 3} d x} (- \frac{\sin x}{x^{3}}) d x + C}{x^{- 3}} c) \ln x - x^{2} y = x y^{'} To determine if it's linear or separable, we can rearrange the equation as x y^{'} - y = \ln x - x^{2} y . Now we see that it's linear. Applying an integrating factor method, the solution is: y (x) = \frac{\int x e^{\int - x d x} (\ln x - x^{2}) d x + C}{x} d) \frac{d y}{d x} + \cos y = \tan x Rearranging the equation, we have \frac{d y}{d x} = \tan x - \cos y . This equation is neither linear nor separable because both y and x appear together in the trigonometric terms. Therefore, the equation is neither linear nor separable. This is helpful 0 FlagDo you have a similar question? Recalculate according to your conditions! RecalculateNew Questions in Linear algebra An object moving in the xy-plane is acted on by a conservative force described by the potential energy function U (x, y) = α (\frac{1}{x^{2}} + \frac{1}{y^{2}}) where α is a positive constant. Derivative an expression for the force expressed terms of the unit vectors \vec{i} and \vec{j} . I need to find a unique description of Nul A, namely by listing the vectors that measure the null space A = [\begin{matrix} 1 & 5 & - 4 & - 3 & 1 \\ 0 & 1 & - 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] ? T must be a linear transformation, we assume. Can u find the T standard matrix. T : R^{2} \to R^{4}, T (e_{1}) = (3, 1, 3, 1) and T (e_{2}) = (- 5, 2, 0, 0), w h e r e e_{1} = (1, 0) and e_{2} = (0, 1) ? 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style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D454 TEX-I\"><\/mjx-c><\/mjx-mi><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-mtext class=\"mjx-n\" space=\"2\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D453 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>g<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mrow><\/mrow><mo>,<\/mo><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi>f<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D454 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>g<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> are known functions.<br>A first-order differntial equation is said to be separable if it can be written as<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45D TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-mtext class=\"mjx-n\" space=\"2\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45D TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>p<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>q<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>,<\/mo><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi>p<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>q<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> are continuouse functions.<br>a) Consider the differential equation <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"3\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D452 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>+<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>Rewrite the given differential equation as follows:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"3\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D452 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mo>\u2212<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-sop\"><mjx-c class=\"mjx-c28 TEX-S1\"><\/mjx-c><\/mjx-mo><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"3\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D452 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-sop\"><mjx-c class=\"mjx-c29 TEX-S1\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo>\u2212<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msup><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>As the variables <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> cannot be separated and written as separate functions of <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>, given equation is not separable.<br>Also, in the equation <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"3\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D452 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-mtext class=\"mjx-n\" space=\"2\"><mjx-c class=\"mjx-cA0\"><\/mjx-c><\/mjx-mtext><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45D TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D452 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>+<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mo>,<\/mo><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi>p<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> but right hand side is not a function of x alone. Hence, it is not a linear equation.<br>Therefore, the given differential equation is neither a separable equation nor a linear equation.<br>b) Consider the differential equation <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-mo class=\"mjx-var\" size=\"s\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>Rewrite the given differential equation as follows:<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-mo class=\"mjx-var\" size=\"s\"><mjx-c class=\"mjx-c2032\"><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msup><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-msup space=\"4\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.413em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>Thus, the given equation can be written as <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c28 TEX-S3\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c29 TEX-S3\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><\/mfrac><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> where<br><span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45D TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2212\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>p<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D45E TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>q<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><\/mrow><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span><br>Hence, it is a linear equation.<br>Also, equation can be written as <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-TeXAtom texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D451 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" style=\"vertical-align: 0.25em\"><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"4\" texclass=\"ORD\"><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-TeXAtom texclass=\"ORD\"><mjx-msup><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-script style=\"vertical-align: 0.289em\"><mjx-TeXAtom size=\"s\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c33\"><\/mjx-c><\/mjx-mn><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-script><\/mjx-msup><\/mjx-TeXAtom><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-TeXAtom><mjx-TeXAtom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"3\" texclass=\"ORD\"><mjx-mi class=\"mjx-n\"><mjx-c class=\"mjx-c73\"><\/mjx-c><mjx-c class=\"mjx-c69\"><\/mjx-c><mjx-c class=\"mjx-c6E\"><\/mjx-c><\/mjx-mi><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2061\"><\/mjx-c><\/mjx-mo><mjx-TeXAtom space=\"2\" texclass=\"ORD\"><mjx-TeXAtom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-TeXAtom><\/mjx-TeXAtom><\/mjx-TeXAtom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-TeXAtom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><mrow data-mjx-texclass=\"ORD\"><mi>d<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\" fence=\"true\" stretchy=\"true\" symmetric=\"true\"><\/mo><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\" fence=\"true\" str","createdAt":"2021-09-21","user":{"id":314,"firstName":"Talisha","lastName":null,"fullName":"Talisha ","shortName":"Talisha ","email":"talisha@email.com","questionsCount":0,"answersCount":93,"avatar":{"id":5,"title":"Female 4","src":"\/build\/images\/user-picture\/user-5.png","src2x":"\/build\/images\/user-picture\/user-5@2x.png"},"level":"Skilled","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"upVotes":30,"questionId":28929,"question":null,"isApproved":true,"isDeleted":false},{"id":118026,"content":"<div class=\"raw-html-embed\">Step 1:<br>a) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><msup><mi>e<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><\/msup><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><msup><mi>y<\/mi><mn>2<\/mn><\/msup><\/mrow><\/math><br>To determine if the equation is linear or separable, we need to rewrite it in a standard form. In this case, we observe that the equation is neither linear nor separable, as the terms involving <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>y<\/mi><\/mrow><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math> cannot be separated.<br>Step 2:<br>b) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>y<\/mi><mo>+<\/mo><mi>sin<\/mi><mi>x<\/mi><mo>=<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math><br>To check linearity, we need to express the equation in the form <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>+<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>y<\/mi><mo>=<\/mo><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>. We can rewrite the given equation as:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mi>sin<\/mi><mi>x<\/mi><\/mrow><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><\/mrow><\/math><br>The equation is linear, as it can be expressed in the desired form.<br>Step 3:<br>c) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>ln<\/mi><mi>x<\/mi><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math><br>To determine if the equation is linear or separable, we need to rewrite it in a standard form. Rearranging the terms, we have:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>x<\/mi><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>y<\/mi><mo>=<\/mo><mi>ln<\/mi><mi>x<\/mi><\/mrow><\/math><br>The equation is linear since it can be written in the form <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>+<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>y<\/mi><mo>=<\/mo><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>.<br>Step 4:<br>d) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>cos<\/mi><mi>y<\/mi><mo>=<\/mo><mi>tan<\/mi><mi>x<\/mi><\/mrow><\/math><br>To check linearity, we need to express the equation in the form <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>+<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>y<\/mi><mo>=<\/mo><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>. We can rewrite the given equation as:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>cos<\/mi><mi>y<\/mi><mo>\u2212<\/mo><mi>tan<\/mi><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/math><br><\/div>","createdAt":"2023-05-13","user":{"id":34413,"firstName":"alenahelenash","lastName":null,"fullName":"alenahelenash ","shortName":"alenahelenash ","email":"alenahelenash@gmail.com","questionsCount":1,"answersCount":556,"avatar":{"id":4,"title":"Female 3","src":"\/build\/images\/user-picture\/user-4.png","src2x":"\/build\/images\/user-picture\/user-4@2x.png"},"level":"Expert","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"upVotes":0,"questionId":28929,"question":null,"isApproved":true,"isDeleted":false},{"id":118027,"content":"<div class=\"raw-html-embed\">Answer:<br>(a)equation neither linear nor separable<br>(b)equation is linear<br>(c)equation is separable<br>(d)equation is linear<br>a) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><msup><mi>e<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><\/msup><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><msup><mi>y<\/mi><mn>2<\/mn><\/msup><\/mrow><\/math><br>To determine if this equation is linear or separable, we need to rewrite it in a standard form. Rearranging the terms, we get:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>e<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><\/msup><\/mrow><\/math><br>This equation is neither linear nor separable since it cannot be expressed in the form <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>g<\/mi><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math> or <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mi>y<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><\/math>.<br>b) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>y<\/mi><mo>+<\/mo><mi>sin<\/mi><mi>x<\/mi><mo>=<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math><br>Rearranging the terms, we have:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>sin<\/mi><mi>x<\/mi><\/mrow><\/math><br>This equation is linear since it can be expressed in the form <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>+<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>y<\/mi><mo>=<\/mo><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>. Here, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><\/mrow><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>sin<\/mi><mi>x<\/mi><\/mrow><\/math>.<br>c) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>ln<\/mi><mi>x<\/mi><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math><br>Rearranging the terms, we obtain:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>x<\/mi><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>y<\/mi><mo>=<\/mo><mi>ln<\/mi><mi>x<\/mi><\/mrow><\/math><br>This equation is separable since we can rewrite it in the form <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>ln<\/mi><mi>x<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mrow><mi>x<\/mi><\/mrow><mrow><mi>y<\/mi><\/mrow><\/mfrac><\/mrow><\/math>.<br>d) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>cos<\/mi><mi>y<\/mi><mo>=<\/mo><mi>tan<\/mi><mi>x<\/mi><\/mrow><\/math><br>This equation is linear since it can be expressed in the form <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>y<\/mi><mo>=<\/mo><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>. Here, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>cos<\/mi><mi>y<\/mi><\/mrow><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>Q<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>tan<\/mi><mi>x<\/mi><\/mrow><\/math>.<br><\/div>","createdAt":"2023-05-13","user":{"id":35119,"firstName":"xleb123","lastName":null,"fullName":"xleb123 ","shortName":"xleb123 ","email":"genshin.sosati@gmail.com","questionsCount":0,"answersCount":181,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Skilled","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"upVotes":0,"questionId":28929,"question":null,"isApproved":true,"isDeleted":false},{"id":118028,"content":"<div class=\"raw-html-embed\">a) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><msup><mi>e<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><\/msup><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><msup><mi>y<\/mi><mn>2<\/mn><\/msup><\/mrow><\/math><br>To determine if this equation is linear or separable, we need to check its form. The equation is neither linear nor separable because the term <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>e<\/mi><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><\/msup><\/mrow><\/math> appears. Therefore, the equation is neither linear nor separable.<br>b) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>y<\/mi><mo>+<\/mo><mi>sin<\/mi><mi>x<\/mi><mo>=<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math><br>Rearranging the equation, we have <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>sin<\/mi><mi>x<\/mi><\/mrow><\/math>. To check if it's linear or separable, we need to see if we can isolate <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math> or <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>y<\/mi><\/mrow><\/math> on one side. Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math> appears with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><\/math>, the equation is not separable. However, we can rewrite it as:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>\u2212<\/mo><mfrac><mrow><mi>y<\/mi><\/mrow><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mrow><mi>sin<\/mi><mi>x<\/mi><\/mrow><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><\/mrow><\/math><br>Now we can see that the equation is linear. Applying an integrating factor method, the solution is:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>y<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo>\u222b<\/mo><msup><mi>x<\/mi><mrow><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/msup><msup><mi>e<\/mi><mrow><mo>\u222b<\/mo><msup><mi>x<\/mi><mrow><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/msup><mi>d<\/mi><mi>x<\/mi><\/mrow><\/msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(<\/mo><mo>\u2212<\/mo><mfrac><mrow><mi>sin<\/mi><mi>x<\/mi><\/mrow><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><\/mfrac><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>d<\/mi><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><\/mrow><mrow><msup><mi>x<\/mi><mrow><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mrow><\/math><br>c) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>ln<\/mi><mi>x<\/mi><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><\/mrow><\/math><br>To determine if it's linear or separable, we can rearrange the equation as <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>x<\/mi><msup><mi>y<\/mi><mi>\u2032<\/mi><\/msup><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mi>ln<\/mi><mi>x<\/mi><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mi>y<\/mi><\/mrow><\/math>. Now we see that it's linear. Applying an integrating factor method, the solution is:<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>y<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo>\u222b<\/mo><mi>x<\/mi><msup><mi>e<\/mi><mrow><mo>\u222b<\/mo><mo>\u2212<\/mo><mi>x<\/mi><mi>d<\/mi><mi>x<\/mi><\/mrow><\/msup><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(<\/mo><mi>ln<\/mi><mi>x<\/mi><mo>\u2212<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>d<\/mi><mi>x<\/mi><mo>+<\/mo><mi>C<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><\/math><br>d) <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>+<\/mo><mi>cos<\/mi><mi>y<\/mi><mo>=<\/mo><mi>tan<\/mi><mi>x<\/mi><\/mrow><\/math><br>Rearranging the equation, we have <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mfrac><mrow><mi>d<\/mi><mi>y<\/mi><\/mrow><mrow><mi>d<\/mi><mi>x<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mi>tan<\/mi><mi>x<\/mi><mo>\u2212<\/mo><mi>cos<\/mi><mi>y<\/mi><\/mrow><\/math>. This equation is neither linear nor separable because both <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>y<\/mi><\/mrow><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\"><mrow><mi>x<\/mi><\/mrow><\/math> appear together in the trigonometric terms. Therefore, the equation is neither linear nor separable.<br><\/div>","createdAt":"2023-05-13","user":{"id":34412,"firstName":"star233","lastName":null,"fullName":"star233 ","shortName":"star233 ","email":"sunny.nikulya.hasp@gmail.com","questionsCount":0,"answersCount":403,"avatar":{"id":5,"title":"Female 4","src":"\/build\/images\/user-picture\/user-5.png","src2x":"\/build\/images\/user-picture\/user-5@2x.png"},"level":"Skilled","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"upVotes":0,"questionId":28929,"question":null,"isApproved":true,"isDeleted":false}],"newQuestions":[{"id":104373,"h1":"An object moving in the xy-plane is acted on by a conservative force described b","content":"An object moving in the xy-plane is acted on by a conservative force described by the potential energy function <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-texatom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D448 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-texatom><mjx-texatom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c28\"><\/mjx-c><\/mjx-mo><mjx-texatom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-texatom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c2C\"><\/mjx-c><\/mjx-mo><mjx-texatom space=\"2\" texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-texatom><mjx-mo class=\"mjx-n\"><mjx-c class=\"mjx-c29\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><mjx-mo class=\"mjx-n\" space=\"4\"><mjx-c class=\"mjx-c3D\"><\/mjx-c><\/mjx-mo><mjx-mi class=\"mjx-i\" space=\"4\"><mjx-c class=\"mjx-c1D6FC TEX-I\"><\/mjx-c><\/mjx-mi><mjx-texatom texclass=\"ORD\"><mjx-mrow><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c28 TEX-S3\"><\/mjx-c><\/mjx-mo><mjx-mfrac><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-texatom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-msup><mjx-texatom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D465 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-texatom><mjx-script style=\"vertical-align: 0.289em\"><mjx-texatom size=\"s\" texclass=\"ORD\"><mjx-texatom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-mo class=\"mjx-n\" space=\"3\"><mjx-c class=\"mjx-c2B\"><\/mjx-c><\/mjx-mo><mjx-mfrac space=\"3\"><mjx-frac type=\"d\"><mjx-num><mjx-nstrut type=\"d\"><\/mjx-nstrut><mjx-texatom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c31\"><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-num><mjx-dbox><mjx-dtable><mjx-line type=\"d\"><\/mjx-line><mjx-row><mjx-den><mjx-dstrut type=\"d\"><\/mjx-dstrut><mjx-msup><mjx-texatom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D466 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-texatom><mjx-script style=\"vertical-align: 0.289em\"><mjx-texatom size=\"s\" texclass=\"ORD\"><mjx-texatom texclass=\"ORD\"><mjx-mn class=\"mjx-n\"><mjx-c class=\"mjx-c32\"><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><mjx-mo class=\"mjx-s3\"><mjx-c class=\"mjx-c29 TEX-S3\"><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>U<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mi>\u03b1<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mfrac><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/msup><\/mfrac><mo>+<\/mo><mfrac><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/msup><\/mfrac><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> where <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D6FC TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mi>\u03b1<\/mi><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> is a positive constant. Derivative an expression for the force expressed terms of the unit vectors <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-texatom texclass=\"ORD\"><mjx-mover><mjx-over style=\"padding-bottom: 0.105em; padding-left: 0.172em; margin-bottom: -0.516em\"><mjx-mo class=\"mjx-n\" style=\"width: 0; margin-left: -0.25em\"><mjx-c class=\"mjx-c20D7 TEX-V\"><\/mjx-c><\/mjx-mo><\/mjx-over><mjx-base><mjx-texatom texclass=\"ORD\"><mjx-texatom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D456 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mover><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><\/mrow><\/mrow><mo stretchy=\"false\">\u2192<\/mo><\/mover><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span> and <span class=\"math-tex\"><mjx-container class=\"MathJax\" jax=\"CHTML\" style=\"font-size: 113.1%; position: relative\"><mjx-math class=\" MJX-TEX\" aria-hidden=\"true\"><mjx-mstyle><mjx-texatom texclass=\"ORD\"><mjx-mover><mjx-over style=\"padding-bottom: 0.105em; padding-left: 0.206em; margin-bottom: -0.516em\"><mjx-mo class=\"mjx-n\" style=\"width: 0; margin-left: -0.25em\"><mjx-c class=\"mjx-c20D7 TEX-V\"><\/mjx-c><\/mjx-mo><\/mjx-over><mjx-base><mjx-texatom texclass=\"ORD\"><mjx-texatom texclass=\"ORD\"><mjx-mi class=\"mjx-i\"><mjx-c class=\"mjx-c1D457 TEX-I\"><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mover><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi>j<\/mi><\/mrow><\/mrow><mo stretchy=\"false\">\u2192<\/mo><\/mover><\/mrow><\/mstyle><\/math><\/mjx-assistive-mml><\/mjx-container><\/span>.\n","category":{"id":38,"name":"Linear algebra","title":"Simplifying Linear Algebra: Techniques, Solutions, and Examples","url":"\/post-secondary\/algebra\/linear-algebra","alias":"linear-algebra","seoText":"Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. Since it is used not only by engineering students but by anyone who has to work with specific calculations, we have provided you with a plethora of questions and answers in their original form. It will help you to see some logic as you are <a href=\"\/secondary\/algebra\/algebra-ii\/complex-numbers\">solving complex numbers<\/a> and understand the basic concepts of linear Algebra in a clearer way. If you need additional help or would like to connect several solutions, compare more than one solution as you approach your task.","parent":{"id":37,"name":"Algebra","title":"College Algebra Solutions for Your Success","url":"\/post-secondary\/algebra","alias":"algebra","seoText":"<div><br>As you are looking for college Algebra solutions, it is vital to receive professional help at the same time because if there are certain mistakes, you will never find reliable solutions that will work for you. <\/div><div><br>Take your time to explore the list of questions and various equations that we have provided along with the answers. As you are told to implement Algebra for your college task, explore the example that we have below.<br><br><\/div>","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-04-01","isAnswered":true,"url":"\/linear-algebra\/104373-an-object-moving-in-the-xy-pla","absoluteUrl":"https:\/\/plainmath.org\/linear-algebra\/104373-an-object-moving-in-the-xy-pla","user":{"id":98581,"firstName":"Annabel Zamora","lastName":"","fullName":"Annabel Zamora ","shortName":"Annabel Zamora ","email":"annabel_zamora_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":1,"title":"Female 1","src":"\/build\/images\/user-picture\/user-1.png","src2x":"\/build\/images\/user-picture\/user-1@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104372,"h1":"Find an explicit description of Nul A by listing vectors that span the null spac","content":"<p>I need to find a unique description of Nul A, namely by listing the vectors that measure the null space<br><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mn>5<\/mn><\/mtd><mtd><mo>\u2212<\/mo><mn>4<\/mn><\/mtd><mtd><mo>\u2212<\/mo><mn>3<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mo>\u2212<\/mo><mn>2<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/math><\/p>?\n","category":{"id":38,"name":"Linear algebra","title":"Simplifying Linear Algebra: Techniques, Solutions, and Examples","url":"\/post-secondary\/algebra\/linear-algebra","alias":"linear-algebra","seoText":"Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. Since it is used not only by engineering students but by anyone who has to work with specific calculations, we have provided you with a plethora of questions and answers in their original form. It will help you to see some logic as you are <a href=\"\/secondary\/algebra\/algebra-ii\/complex-numbers\">solving complex numbers<\/a> and understand the basic concepts of linear Algebra in a clearer way. If you need additional help or would like to connect several solutions, compare more than one solution as you approach your task.","parent":{"id":37,"name":"Algebra","title":"College Algebra Solutions for Your Success","url":"\/post-secondary\/algebra","alias":"algebra","seoText":"<div><br>As you are looking for college Algebra solutions, it is vital to receive professional help at the same time because if there are certain mistakes, you will never find reliable solutions that will work for you. <\/div><div><br>Take your time to explore the list of questions and various equations that we have provided along with the answers. As you are told to implement Algebra for your college task, explore the example that we have below.<br><br><\/div>","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-31","isAnswered":true,"url":"\/linear-algebra\/104372-find-an-explicit-description-o","absoluteUrl":"https:\/\/plainmath.org\/linear-algebra\/104372-find-an-explicit-description-o","user":{"id":98358,"firstName":"Oswaldo Riley","lastName":"","fullName":"Oswaldo Riley ","shortName":"Oswaldo Riley ","email":"oswaldo_riley_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":2,"title":"Male 1","src":"\/build\/images\/user-picture\/user-2.png","src2x":"\/build\/images\/user-picture\/user-2@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104371,"h1":"Assume that T is a linear transformation. Find the standard matrix of T. T:\\m","content":"<p>T must be a linear transformation, we assume. Can u find the T standard matrix.<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mi>T<\/mi><\/mrow><mo>:<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">R<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mo stretchy=\"false\">\u2192<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">R<\/mi><\/mrow><\/mrow><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>4<\/mn><\/mrow><\/mrow><\/mrow><\/msup><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>T<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>3<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mstyle scriptlevel=\"0\"><mspace width=\"1em\">\u00a0<\/mspace><\/mstyle><mtext>and<\/mtext><mstyle scriptlevel=\"0\"><mspace width=\"1em\">\u00a0<\/mspace><\/mstyle><\/mrow><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi>T<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>5<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mo>,<\/mo><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi>w<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>h<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>r<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mtext>\u00a0<\/mtext><msub><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><mtext>\u00a0<\/mtext><mrow data-mjx-texclass=\"ORD\"><mstyle scriptlevel=\"0\"><mspace width=\"1em\">\u00a0<\/mspace><\/mstyle><mtext>and<\/mtext><mstyle scriptlevel=\"0\"><mspace width=\"1em\">\u00a0<\/mspace><\/mstyle><\/mrow><mtext>\u00a0<\/mtext><msub><mrow data-mjx-texclass=\"ORD\"><mi>e<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mn>2<\/mn><\/mrow><\/mrow><\/mrow><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mn>1<\/mn><\/mrow><mo data-mjx-texclass=\"CLOSE\">)<\/mo><\/mrow><\/mrow><\/mstyle><\/math><\/p>?\n","category":{"id":38,"name":"Linear algebra","title":"Simplifying Linear Algebra: Techniques, Solutions, and Examples","url":"\/post-secondary\/algebra\/linear-algebra","alias":"linear-algebra","seoText":"Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. Since it is used not only by engineering students but by anyone who has to work with specific calculations, we have provided you with a plethora of questions and answers in their original form. It will help you to see some logic as you are <a href=\"\/secondary\/algebra\/algebra-ii\/complex-numbers\">solving complex numbers<\/a> and understand the basic concepts of linear Algebra in a clearer way. If you need additional help or would like to connect several solutions, compare more than one solution as you approach your task.","parent":{"id":37,"name":"Algebra","title":"College Algebra Solutions for Your Success","url":"\/post-secondary\/algebra","alias":"algebra","seoText":"<div><br>As you are looking for college Algebra solutions, it is vital to receive professional help at the same time because if there are certain mistakes, you will never find reliable solutions that will work for you. <\/div><div><br>Take your time to explore the list of questions and various equations that we have provided along with the answers. As you are told to implement Algebra for your college task, explore the example that we have below.<br><br><\/div>","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-30","isAnswered":true,"url":"\/linear-algebra\/104371-assume-that-t-is-a-linear-tran","absoluteUrl":"https:\/\/plainmath.org\/linear-algebra\/104371-assume-that-t-is-a-linear-tran","user":{"id":98408,"firstName":"Aron Campbell","lastName":"","fullName":"Aron Campbell ","shortName":"Aron Campbell ","email":"aron_campbell_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":7,"title":"Male 3","src":"\/build\/images\/user-picture\/user-7.png","src2x":"\/build\/images\/user-picture\/user-7@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104363,"h1":"Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR\n\nConsider the points below\n\nP(1,0,1)  , Q(-2,1,4) , R(7,2,7)\n\na) Find a nonzero vector orthogonal to the plane through the points P,Q and R\n\nb) Find the area of the triangle PQR","content":"Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR<br>Consider the points below<br>P(1,0,1)  , Q(-2,1,4) , R(7,2,7).<br>a) Find a nonzero vector orthogonal to the plane through the points P,Q and R.<br>b) Find the area of the triangle PQR.\n","category":{"id":38,"name":"Linear algebra","title":"Simplifying Linear Algebra: Techniques, Solutions, and Examples","url":"\/post-secondary\/algebra\/linear-algebra","alias":"linear-algebra","seoText":"Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. Since it is used not only by engineering students but by anyone who has to work with specific calculations, we have provided you with a plethora of questions and answers in their original form. It will help you to see some logic as you are <a href=\"\/secondary\/algebra\/algebra-ii\/complex-numbers\">solving complex numbers<\/a> and understand the basic concepts of linear Algebra in a clearer way. If you need additional help or would like to connect several solutions, compare more than one solution as you approach your task.","parent":{"id":37,"name":"Algebra","title":"College Algebra Solutions for Your Success","url":"\/post-secondary\/algebra","alias":"algebra","seoText":"<div><br>As you are looking for college Algebra solutions, it is vital to receive professional help at the same time because if there are certain mistakes, you will never find reliable solutions that will work for you. <\/div><div><br>Take your time to explore the list of questions and various equations that we have provided along with the answers. As you are told to implement Algebra for your college task, explore the example that we have below.<br><br><\/div>","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-30","isAnswered":true,"url":"\/linear-algebra\/104363-find-a-nonzero-vector-orthogon","absoluteUrl":"https:\/\/plainmath.org\/linear-algebra\/104363-find-a-nonzero-vector-orthogon","user":{"id":98471,"firstName":"Jacoby Blevins","lastName":"","fullName":"Jacoby Blevins ","shortName":"Jacoby Blevins ","email":"jacoby_blevins_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":3,"title":"Female 2","src":"\/build\/images\/user-picture\/user-3.png","src2x":"\/build\/images\/user-picture\/user-3@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104093,"h1":"Consider two vectors  A=3i  - 1j  and B = - i  - 5j, how do you calculate  A - B?","content":"Consider two vectors  A=3i  - 1j  and B = - i  - 5j, how do you calculate  A - B?\n","category":{"id":38,"name":"Linear algebra","title":"Simplifying Linear Algebra: Techniques, Solutions, and Examples","url":"\/post-secondary\/algebra\/linear-algebra","alias":"linear-algebra","seoText":"Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. Since it is used not only by engineering students but by anyone who has to work with specific calculations, we have provided you with a plethora of questions and answers in their original form. It will help you to see some logic as you are <a href=\"\/secondary\/algebra\/algebra-ii\/complex-numbers\">solving complex numbers<\/a> and understand the basic concepts of linear Algebra in a clearer way. If you need additional help or would like to connect several solutions, compare more than one solution as you approach your task.","parent":{"id":37,"name":"Algebra","title":"College Algebra Solutions for Your Success","url":"\/post-secondary\/algebra","alias":"algebra","seoText":"<div><br>As you are looking for college Algebra solutions, it is vital to receive professional help at the same time because if there are certain mistakes, you will never find reliable solutions that will work for you. <\/div><div><br>Take your time to explore the list of questions and various equations that we have provided along with the answers. As you are told to implement Algebra for your college task, explore the example that we have below.<br><br><\/div>","parent":{"id":36,"name":"College","title":"College Math Questions: Expert Solutions & Tips","url":"\/post-secondary","alias":"post-secondary","seoText":"College Math problems only seem challenging at first, yet what they truly represent is not much harder than high school Math with some equations that are closer to the engineering approach. Approaching your college math helper is essential because your questions may just have to be worded differently. Regardless if you are dealing with equations or need answers to your college math questions, taking a look at college math equations will let you see things differently. Approach college math answers step-by-step. 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The majority of college math problems with answers represented here will be similar to what you already have.","parent":null,"children":[]},"children":[]},"children":[]},"createdAt":"2023-03-23","isAnswered":true,"url":"\/linear-algebra\/103931-what-is-the-projection-of-6-5","absoluteUrl":"https:\/\/plainmath.org\/linear-algebra\/103931-what-is-the-projection-of-6-5","user":{"id":98566,"firstName":"miljanwpd2","lastName":"","fullName":"miljanwpd2 ","shortName":"miljanwpd2 ","email":"miljanwpd2_@email.com","questionsCount":0,"answersCount":0,"avatar":{"id":6,"title":"Male 2","src":"\/build\/images\/user-picture\/user-6.png","src2x":"\/build\/images\/user-picture\/user-6@2x.png"},"level":"Beginner","freeQuestionsCount":0,"info":null,"subscription":null,"subscriptionService":null,"snInfo":null},"video":"","isPrivate":false,"isPublished":true,"isHidden":false,"answersCount":1,"isWaitingForPayment":false,"isPaid":false,"isApproved":true,"isDeleted":false,"payUrl":null},{"id":104109,"h1":"What is the dot product of #<1,-4,5   ># and #<-5,7,3   >#?","content":"What is the dot product of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">  <mstyle displaystyle=\"true\">    <mo><<\/mo>    <mn>1<\/mn>    <mo>,<\/mo>    <mo>-<\/mo>    <mn>4<\/mn>    <mo>,<\/mo>    <mn>5<\/mn>    <mo>><\/mo>  <\/mstyle><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">  <mstyle displaystyle=\"true\">    <mo><<\/mo>    <mo>-<\/mo>    <mn>5<\/mn>    <mo>,<\/mo>    <mn>7<\/mn>    <mo>,<\/mo>    <mn>3<\/mn>    <mo>><\/mo>  <\/mstyle><\/math>?\n","category":{"id":38,"name":"Linear algebra","title":"Simplifying Linear Algebra: Techniques, Solutions, and Examples","url":"\/post-secondary\/algebra\/linear-algebra","alias":"linear-algebra","seoText":"Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. 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