Chapter 5.7, Problem 25P

a) By the method of variation of parameters, show that the solution of the initial value problem

$y\text{'}\text{'}+2y\text{'}+2y=f(t),$ $y(0)=0,$ $y\text{'}(0)=0$

Is

$y={\displaystyle \underset{0}{\overset{t}{\int }}{e}^{-(t-\tau )}f(\tau )\sin (t-\tau )d\tau .}$

b) Show that if $f(t)=\delta (t-\pi ),$ then the solution of part (a) reduces to

$y=u_{\pi }(t)e^{-(t-\pi )}\sin (t-\pi ).$

c) Use a Laplace transform to solve the given initial value problem with $f(t)=\delta (t-\pi )$ and confirm that the solution agrees with the result of part (b).