91Ó°ÊÓ

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

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

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

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 \(\Delta t\). $$y^{\prime}=\cos (t+y), y(0)=1$$

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

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

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

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

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 \(\Delta t\). $$y^{\prime}=t \ln y, y(0)=2$$