Chapter 4.7, Problem 35P

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

$L[y]=\left(D-a\right)\left(D-b\right)y=g\left(t\right)$,

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

where $a$ and $b$ are real numbers with $a\ne b$

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)