Chapter 4.7, Problem 36P

Use the result of Problem 33 to find the solution of the initial value problem

$L[y]=[D^2-2\lambda D+\left({\lambda }^2+{\mu }^2\right)]y=g\left(t\right)$,

$y\left(t_0\right)=0,y\text{'}\left(t_0\right)=0$

Note that the roots of the characteristic equation are $\lambda \pm \mu i$.

33. By choosing the lower limit of integration in Eq. (27) in the text as the initial point $t_0$, show that $Y(t)$ becomes

$Y\left(t\right)={\displaystyle \underset{t_0}{\overset{t}{\int }}\frac{y_1\left(\tau \right)y_2\left(t\right)-y_1\left(t\right)y_2\left(\tau \right)}{y_1\left(\tau \right)y_2\text{'}\left(\tau \right)-y_1\text{'}\left(\tau \right)y_2\left(\tau \right)}}g\left(\tau \right)d\tau$

Show that $Y(t)$ is a solution of the initial value problem

$L[y]=g\left(t\right)$,      $y\left(t_0\right)=0$,      $y\text{'}\left(t_0\right)=0$

Thus $Y$ can be identified with $v$ in problem 32.

$Y\left(t\right)=-y_1\left(t\right){\displaystyle \int \frac{y_2\left(t\right)g\left(t\right)}{W[y_1,y_2]\left(t\right)}}dt+y_2\left(t\right){\displaystyle \int \frac{y_1\left(t\right)g\left(t\right)}{W[y_1,y_2]\left(t\right)}}dt$ (27)