Existence of solution for second-order linear differential equations

Question

It is "widely known" that the initial value problem
$$y''+p(t)y'+q(t)y=g(t),\qquad y(t_0)=y_0,\ \ y'(t_0)=y'_0$$
where $p(t),q(t),g(t)$ are continuous in an open interval $I$, has a unique solution in $I$, but few textbooks present the proof, or even give a sketch of the proof. After some googling, I found that the uniqueness can be shown relatively easily (albeit nontrivially). My question is, how do you prove the existence? At least, what is the idea? (Converting it to a system of first-order equations and applying the Picard iteration does not seem to guarantee that the solution exists over the entire $I$.)

Answer 1

It can be converted to a system of first order linear equations by letting $z=y'$. $$\begin{pmatrix}y'\\z'\end{pmatrix}=\begin{pmatrix}0&1\\-q&-p\end{pmatrix}\begin{pmatrix}y\\z\end{pmatrix}+\begin{pmatrix}0\\g\end{pmatrix},\qquad\begin{pmatrix}y(t_0)\\z(t_0)\end{pmatrix}=\begin{pmatrix}y_0\\y_0'\end{pmatrix}.$$ The existence of a particular solution of such a system can be established in a number of ways, for example by the method of variation of parameters, according to which $$\begin{pmatrix}y\\z\end{pmatrix}=\Phi\int\Phi^{-1}\begin{pmatrix}0\\g\end{pmatrix},$$ where $\Phi$ is the fundamental matrix.