3
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

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 $x,y$.

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – Dac0 Aug 5 '18 at 16:09
  • $\begingroup$ Can you show me why is effectively Lipschitz? $\endgroup$ – Dac0 Aug 5 '18 at 16:16
  • $\begingroup$ It is linear in $x$ and $y$. $\endgroup$ – Julián Aguirre Aug 5 '18 at 18:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.