text: Q:
  
  Existence and uniqueness for second order ODE
  
  Given a second-order ODE of the form $$x''(t) = f(t,x(t),x'(t))$$
  with initial conditions $x(t_0) = x_0 \in \mathbb{R}^n$ and $x'(t_0) = y_0 \in \mathbb{R}^n$ for $t_0 \in \mathbb{R}$
  What theorem can be used to prove the existence and uniqueness of solutions to this general equation? 
  I only know Picard-Lindelöf and the resources I find online are almost all computational. Thanks.
  
  A:
  
  You convert the problem into the first order system $x'=y,y'=f(t,x,y)$, and then apply a theorem such as Picard-Lindelöf for first order systems.
  
  