# Existence and uniqueness of 2nd order linear differential equations

I know that the equation $$\frac{d^{2}x}{dt^{2}}+p\left(t\right)\frac{dx}{dt}+q\left(t\right)x=g\left(t\right),$$ has a unique solution on open sets where $p\left(t\right),q\left(t\right)$ and $g\left(t\right)$ are continuous. What I was wondering if this fact could be derived from the Picard's Theorem on Uniqueness and Existence of First ODE making the usual substitution $y=x'$ and $y_0=x(t_0)$. If so, why $p\left(t\right),q\left(t\right)$ do not need to be Lipschitz and only need to be continuous?

## 1 Answer

In Picard's theorem for a system $y'(t)=F(t,y)$, $F$ must be continuous in both variables and locally Lipschitz in the $y$ variable. The second order differential equation is equivalent to the system \begin{align} x'&=y\\ y'&=-p(t)\,y-q(t)\,x+g(t) \end{align} The right hand side is continuous in both variables and Lipschitz in $x,y$.

• So you are positive that the existance and uniqueness of 2nd order linear function can be derived from the Picard's Theorem on Uniqueness and Existence of first order ODE? – Dac0 Aug 5 '18 at 16:09
• Can you show me why is effectively Lipschitz? – Dac0 Aug 5 '18 at 16:16
• It is linear in $x$ and $y$. – Julián Aguirre Aug 5 '18 at 18:51