4.2.1 Existence and uniqueness for linear second order ode

Given linear second order ode

\[ y^{\prime \prime }+p\left ( x\right ) y^{\prime }+q\left ( x\right ) y=f\left ( x\right ) \]

With initial conditions at \(x_{0}\)

\begin{align*} y\left ( x_{0}\right ) & =y_{0}\\ y^{\prime }\left ( x_{0}\right ) & =y_{0}^{\prime }\end{align*}

If \(p\left ( x\right ) ,q\left ( x\right ) ,f\left ( x\right ) \) are all continuous at \(x_{0}\) then theorem guarantees that a solution exist and is unique on some interval than includes \(x_{0}\). If this was not the case, (i.e. if any of \(p,q,f\) are not continuous at \(x_{0}\)) then the theorem does not apply. This means a solution could still exists and even be unique, but theory does not say anything about this.