Calculus 2 Equations: Expert Solutions and More

${\displaystyle y^{}+3y^{}+4y=2\cos 2t}$

True or False? Justify your answer with a proof or a counterexample. You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.

Show that the second-order differential equation $y″=F(x,y,y\prime )$ can be reduced to a system of two first-order differential equations
${\displaystyle \frac{dy}{dx}=z,\frac{dz}{dx}=F(x,y,z).}$
Can something similar be done to the nth-order differential equation
${\displaystyle y^{\left(n\right)}=F(x,y,y^{\prime },y{}^{″},\dots ,y^{(n-1)})?}$

Express the plane $z=x$ in cylindrical and spherical coordinates.
a) cykindrical
$z=r\cos (\theta )$
b) spherical coordinates
$\theta =\arcsin (\cot (\varphi ))$

if tax revenue is $230 billion and the governments

If $xy+6e^y=6e$ , find the value of y" at the point where x=0

Use the substitution $v=y^{\prime }$ to write each second-order equation as a system of two first-order differential equations (planar system). $y^{″}+2y^{\prime }-3y=0$

Math the parametric equation with the correct graph

Solve the given differential equation by separation of variables: $\frac{dy}{dx}=x\sqrt{1-y^2}$

Change from rectangular to spherical coordinates. (Let $\rho \ge 0,0\le \theta \le 2\pi$, and $0\le \varphi \le \pi .)$
a) $(0,-8,0)(\rho ,\theta ,\varphi )=?$
b) $(-1,1,-2)(\rho ,\theta ,\varphi )=?$

Use the given graph off over the interval (0, 6) to find the following.

a) The open intervals on whichfis increasing. (Enter your answer using interval notation.)
b) The open intervals on whichfis decreasing. (Enter your answer using interval notation.)
c) The open intervals on whichfis concave upward. (Enter your answer using interval notation.)
d) The open intervals on whichfis concave downward. (Enter your answer using interval notation.)
e) The coordinates of the point of inflection. $(x,y)=$

Use the substitution $v=y^{\prime }$ to write each second-order equation as a system of two first-order differential equations (planar system). $4y^{″}+4y^{\prime }+y=0$

Solve the differential equation by variation of parameters
$y"+3y^{\prime }+2y=\frac{1}{1+e^x}$

Transform the given differential equation or system into an equivalent system of first-order differential equations.
${\displaystyle x^{″}+3x^{\prime }-7x=t^2}$

Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.

Find $\frac{dy}{dx}$ and $\frac{d^{2}y}{d{x}^2}.x=e^t,y=t{e}^{-t}$. For which values of t is the curve concave upward?

Change from rectangular to spherical coordinates. (Let $\rho \ge 0,0\le \theta \le 2\pi ,\text{ and }0\le \varphi \le \pi$)
(a) (0, -3, 0)
(b) $(-1,1,-\sqrt{2})$

Transform the given differential equation or system into an equivalent system of first-order differential equations.
${\displaystyle x^{\left(3\right)}={\left(x^{\prime }\right)}^2+\cos x}$

Use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Test is inconclusive, determine the convergence or divergence of the series using other methods.
$\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{(n+1)(n^2+2)}$

Find the radius of convergence, R, of the series.
$\sum _{n=1}^{\mathrm{\infty }}\frac{(2x+9{)}^n}{n^2}$
Find the interval, I, of convergence of the series.