Chapter 4.7, Problem 32P

Show that the solution of the initial value problem

$L[y]=y\text{'}\text{'}+p\left(t\right)y\text{'}+q\left(t\right)y=g\left(t\right)$,

$y\left(t_0\right)=y_0$,      $y\text{'}\left(t_0\right)=y_1$

can be written as $y=u\left(t\right)+v\left(t\right)$, where $u$ and $v$ are solutions of the two initial value problems

$L[u]=0$, $u\left(t_0\right)=y_0$, $u\text{'}\left(t_0\right)=y_1$,

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

respectively. In other words, the nonhomogeneities in the differential equation and in the initial conditions can be dealt with separately. Observe that $u$ is easy to find if a fundamental set of solutions of $L[u]=0$ is known.