Chapter 10

Find the general solution to the differential equation using variation of parameters. $$y^{\prime \prime}+y=\tan x$$

Solve the initial value problem \(y^{\prime \prime}-\omega^{2} y=0, y(0)=1, y^{\prime}(0)=1,\) assuming \(\omega \neq 0\).

In the following exercises, compute the Euler approximations for the initial value problem for \(0 \leq t \leq 1\) and \(\Delta t=0.2 .\) If you have access to Sage, generate the slope field first and attempt to sketch the solution curve. Then use Sage to compute better approximations with smaller values of \(\Delta t\). $$y^{\prime}=t / y, y(0)=1$$

Find the general solution of the equation. $$y^{\prime}+4 y=8$$

Find the general solution of each equation in the following exercises. $$y^{\prime}+5 y=0$$

Which of the following equations are separable? (a) \(y^{\prime}=\sin (t y)\) (b) \(y^{\prime}=e^{t} e^{y}\) (c) \(y y^{\prime}=t\) (d) \(y^{\prime}=\left(t^{3}-t\right) \arcsin (y)\) (e) \(y^{\prime}=t^{2} \ln y+4 t^{3} \ln y\)

Find the general solution to the differential equation using variation of parameters. $$y^{\prime \prime}+y=e^{2 t}$$

Solve the initial value problem \(2 y^{\prime \prime}+18 y=0, y(0)=2, y^{\prime}(0)=15\).

In the following exercises, compute the Euler approximations for the initial value problem for \(0 \leq t \leq 1\) and \(\Delta t=0.2 .\) If you have access to Sage, generate the slope field first and attempt to sketch the solution curve. Then use Sage to compute better approximations with smaller values of \(\Delta t\). $$y^{\prime}=t+y^{3}, y(0)=1$$

Find the general solution of the equation. $$y^{\prime}-2 y=6$$

Find the general solution of each equation in the following exercises. $$y^{\prime}-2 y=0$$

Solve \(y^{\prime}=1 /\left(1+t^{2}\right)\).

Find the general solution to the differential equation using variation of parameters. $$y^{\prime \prime}+4 y=\sec x$$

Solve the initial value problem \(y^{\prime \prime}+6 y^{\prime}+5 y=0, y(0)=1, y^{\prime}(0)=0\).

In the following exercises, compute the Euler approximations for the initial value problem for \(0 \leq t \leq 1\) and \(\Delta t=0.2 .\) If you have access to Sage, generate the slope field first and attempt to sketch the solution curve. Then use Sage to compute better approximations with smaller values of \(\Delta t\). $$y^{\prime}=\cos (t+y), y(0)=1$$

Find the general solution of the equation. $$y^{\prime}+t y=5 t$$

Find the general solution of each equation in the following exercises. $$y^{\prime}+\frac{y}{1+t^{2}}=0$$

Solve the initial value problem \(y^{\prime}=t^{n}\) with \(y(0)=1\) and \(n \geq 0 .\)

Find the general solution to the differential equation using variation of parameters. $$y^{\prime \prime}+4 y=\tan x$$

Solve the initial value problem \(y^{\prime \prime}-y^{\prime}-12 y=0, y(0)=0, y^{\prime}(0)=14 .\)

In the following exercises, compute the Euler approximations for the initial value problem for \(0 \leq t \leq 1\) and \(\Delta t=0.2 .\) If you have access to Sage, generate the slope field first and attempt to sketch the solution curve. Then use Sage to compute better approximations with smaller values of \(\Delta t\). $$y^{\prime}=t \ln y, y(0)=2$$

Find the general solution of the equation. $$y^{\prime}+e^{t} y=-2 e^{t}$$

Find the general solution of each equation in the following exercises. $$y^{\prime}+t^{2} y=0$$

Solve \(y^{\prime}=\ln t\).

Find the general solution to the differential equation using variation of parameters. $$y^{\prime \prime}+y^{\prime}-6 y=t^{2} e^{2 t}$$

Solve the initial value problem \(y^{\prime \prime}+12 y^{\prime}+36 y=0, y(0)=5, y^{\prime}(0)=-10\).

Find the general solution of the equation. $$y^{\prime}-y=t^{2}$$

Solve the initial value problem. $$y^{\prime}+y=0, y(0)=4$$

Identify the constant solutions (if any) of \(y^{\prime}=t \sin y .\)

Find the general solution to the differential equation using variation of parameters. $$y^{\prime \prime}-2 y^{\prime}+2 y=e^{t} \tan (t)$$

Solve the initial value problem \(y^{\prime \prime}-8 y^{\prime}+16 y=0, y(0)=-3, y^{\prime}(0)=4\).

Find the general solution of the equation. $$2 y^{\prime}+y=t$$

Solve the initial value problem. $$y^{\prime}-3 y=0, y(1)=-2$$

Identify the constant solutions (if any) of \(y^{\prime}=t e^{y}\).

Find the general solution to the differential equation. $$y^{\prime \prime}+y^{\prime}-6 y=e^{-3 t}$$

Solve the initial value problem \(y^{\prime \prime}+5 y=0, y(0)=-2, y^{\prime}(0)=5\).

Find the general solution of the equation. $$t y^{\prime}-2 y=1 / t, t>0$$

Solve the initial value problem. $$y^{\prime}+y \sin t=0, y(\pi)=1$$

Solve \(y^{\prime}=t / y\).

Find the general solution to the differential equation. $$y^{\prime \prime}-4 y^{\prime}+3 y=e^{3 t}$$

Solve the initial value problem \(y^{\prime \prime}+y=0, y(\pi / 4)=0, y^{\prime}(\pi / 4)=2\).

Find the general solution of the equation. $$t y^{\prime}+y=\sqrt{t}, t>0$$

Solve the initial value problem. $$y^{\prime}+y e^{t}=0, y(0)=e$$

Solve \(y^{\prime}=y^{2}-1\).

Find the general solution to the differential equation. $$y^{\prime \prime}+16 y=\cos (4 t)$$

Solve the initial value problem \(y^{\prime \prime}+12 y^{\prime}+37 y=0, y(0)=4, y^{\prime}(0)=0\).

Find the general solution of the equation. $$y^{\prime} \cos t+y \sin t=1,-\pi / 2

Solve the initial value problem. $$y^{\prime}+y \sqrt{1+t^{4}}=0, y(0)=0$$

Solve \(y^{\prime}=t /\left(y^{3}-5\right) .\) You may leave your solution in implicit form: that is, you may stop once you have done the integration, without solving for \(y .\)

Find the general solution to the differential equation. $$y^{\prime \prime}+9 y=3 \sin (3 t)$$