Solve System of Differential Equations with Expert Help

Inverse Laplace transformation

$(s^2+s)/(s^2+1)(s^2+2s+2)$

Write first-order differential equations that express the situation: A student in an engineering course finds the rate of increase of his grade point average GG is directly proportional to the number NN of study hours/week and inversely proportional to the amount AA of time spent on online games .

Which is a narrow, debatable claim?

You have a $150 gift card to use at a sporting goods store. You buy 2 pairs of shoes for $65. You plan to spend the rest of the money on socks. Socks cost $4.75 per pair. What is the greatest number of pairs of socks you can purchase?

Use the table of Laplace transform and properties to obtain the Laplace transform of the following functions. Specify which transform pair or property is used and write in the simplest form.
a) $x(t)=\cos (3t)$
b)$y(t)=t\cos (3t)$
c) $z(t)=e^{-2t}[t\cos (3t)]$
d) $x(t)=3\cos (2t)+5\sin (8t)$
e) $y(t)=t^3+3t^2$
f) $z(t)=t^4{e}^{-2t}$

If $xy+8e^y=8e$ , find the value of y" at the point where $x=0$
$y"=?$

Identify the surface whose equation is given.
$\rho =\sin \theta \sin \varphi$

Solve the differential equation by variation of parameters
$y"+y=\sin x$

The integrating factor method, which was an effective method for solving first-order differential equations, is not a viable approach for solving second-order equstions. To see what happens, even for the simplest equation, consider the differential equation ${\displaystyle y{}^{″}+3y^{\prime }+2y=f\left(t\right)}$. Lagrange sought a function ${\displaystyle \mu \left(t\right)\mu \left(t\right)}$ such that if one multiplied the left-hand side of ${\displaystyle y{}^{″}+3y^{\prime }+2y=f\left(t\right)}$ bu ${\displaystyle \mu \left(t\right)\mu \left(t\right)}$, one would get ${\displaystyle \mu \left(t\right)[y{}^{″}+y^{\prime }+y]=d\frac{d}{dt}[\mu \left(t\right)y+g\left(t\right)y]}$ where g(t)g(t) is to be determined. In this way, the given differential equation would be converted to ${\displaystyle \frac{d}{dt}[\mu \left(t\right)y^{\prime }+g\left(t\right)y]=\mu \left(t\right)f\left(t\right)}$, which could be integrated, giving the first-order equation ${\displaystyle \mu \left(t\right)y^{\prime }+g\left(t\right)y=\int \mu \left(t\right)f\left(t\right)dt+c}$ which could be solved by first-order methods. (a) Differentate the right-hand side of ${\displaystyle \mu \left(t\right)[y{}^{″}+y^{\prime }+y]=d\frac{d}{dt}[\mu \left(t\right)y+g\left(t\right)y]}$ and set the coefficients of y,y' and y'' equal to each other to find g(t). (b) Show that the integrating factor ${\displaystyle \mu \left(t\right)\mu \left(t\right)}$ satisfies the second-order homogeneous equation ${\displaystyle \mu {}^{″}-{\mu }^{\prime }+\mu =0}$ called the adjoint equation of ${\displaystyle y{}^{″}+3y^{\prime }+2y=f\left(t\right)}$. In other words, althought it is possible to find an "integrating factor" for second-order differential equations, to find it one must solve a new second-order equation for the integrating factor μ, which might be every bit as hard as the original equation. (c) Show that the adjoint equation of the general second-order linear equation ${\displaystyle y{}^{″}+p\left(t\right)y^{\prime }+q\left(t\right)y=f\left(t\right)}$ is the homogeneous equation ${\displaystyle \mu {}^{″}-p\left(t\right){\mu }^{\prime }+[q\left(t\right)-p^{\prime }\left(t\right)]\mu =0}$.

Solve the given differential equation by separation of variables:
$\frac{dP}{dt}=P-P^2$

Write first-order differential equations that express the situation: A Walking Mode, l Construct a mathematical model to estimate how long it will take you to walk to the store for groceries. Suppose you have measured the distance as 11 mile and you estimate your walking speed as 33 miles per hour. If it takes you 2020 minutes to get to the store, what do you conclude about your model?

Write the following first-order differential equations in standard form. ${\displaystyle y^{\prime }=x^{3}y+\sin x}$

Find the differential of each function.
(a) $y=\tan \sqrt{t}$
(b) $y=\frac{1-v^2}{1+v^2}$

Write the following first-order differential equations in standard form. ${\displaystyle \frac{dy}{dt}=yx(x+1)}$

A population like that of the United States with an age structure of roughly equal numbers in each of the age groups can be predicted to A) grow rapidly over a 30-year-period and then stabilize B) grow little for a generation and then grow rapidly C) fall slowly and steadily over many decades D) show slow and steady growth for some time into the future

what is the squareroot of 2

Find an equation of the tangent line to the curve at the given point (9, 3)
$y=\frac{1}{\sqrt{x}}$

Use the family in Problem 1 to find a solution of $y+y=0$ that satisfies the boundary conditions $y(0)=0,y(1)=1.$

Find equations of both lines through the point (2, ?3) that are tangent to the parabola $y=x^2+x$.
$y_1$=(smaller slope quation)
$y_2$=(larger slope equation)

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. $f(x)=\left\{\frac{1}{x+2}\right\}$ if $x\ne -2$ $a=-2$
1 if $x=-2$