Chapter 4.7, Problem 38P

By combining the results of the problems $35$ through $37$, show that the solution of the initial value problem

$L[y]=\left(D^2+bD+c\right)y=g\left(t\right),$

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

where $b$ and $c$ are constants, has the form

$y=\varphi \left(t\right)={\displaystyle \underset{t_0}{\overset{t}{\int }}K\left(t-s\right)g\left(s\right)}ds$         (i)

The function $K$ depends only on the solutions $y_1$ and $y_2$ of the corresponding homogeneous equation and is independent of the nonhomogeneous term. Once $K$ is determined, all nonhomogeneous problems involving the same differential operator $L$ are reduced to the evaluation of an integral. Note also that, although $K$ depends on both $t$ and $s$, only the combination $t-s$ appears, so $K$ is actually a function of a single variable. When we think of $g(t)$ as the input to the problem and of $\varphi (t)$ as output, it follows from Eq. (i) that the output depends on the input over the entire interval from the initial point $t_0$ to the current value $t$. The integral in Eq. (i) is called the convolution of $K$ and $g$, and $K$ is referred to as the kernel.