Chapter 4.5, Problem 38P

Follow the instructions in Problem 37 to solve the differential equation

$y\text{'}\text{'}+2y\text{'}+5y=\{\begin{array}{c}1,\\ 0,\end{array}\begin{array}{c}0\le t\le \pi /2,\\ t>\pi /2,\end{array}$

With the initial conditions $y\left(0\right)=0$ and $y\text{'}\left(0\right)=0$.

In many physical problems, the nonhomogeneous term may be specified by different formulas in different time periods. As an example, determine the solution $y=\varphi \left(t\right)$ of

$y\text{'}\text{'}+y=\{\begin{array}{c}t,\\ \pi e^{\pi -t},\end{array}\begin{array}{c}0\le t\le \pi ,\\ t>\pi ,\end{array}$

Satisfying the initial conditions $y\left(0\right)=0$ and $y\text{'}\left(0\right)=1$. Assume that $y$ and $y\text{'}$ are also continuous at $t=\pi$. Plot the nonhomogeneous term and the solution as functions of time.

Hint: First solve the initial value problem for $t\le \pi$; then solve for $t>\pi$, determining the constants in the latter solution from the continuity conditions at $t=\pi$.