Cauchy-Lipschitz Theorem for second order ODEs

Question

How do I verify if Cauchy-Lipschitz Theorem is valid for second order ODEs? For example:

$$\begin{cases} x''-x'-2x=te^{-2t}\\ x(0)=5/16\\ x'(1)=0 \end {cases}$$

I know to do this in first order ODEs by checking if $f(x,t)$ is uniformly Lipschitz continuous (with partial derivatives) but no clue with high order.

Thanks.

Answer 1

Generally for using the Cauchy-Lipschitz theorem, you must convert a non first-order ODE to a first-order one. For example in here by defining $y=x'-2x$ we obtain $$y'+y=te^{-2t}$$