Find an ODE of the form $\displaystyle y''+ay'+by=0$ where $\displaystyle a^2>4b$ for which the given function is a solution, or else explain why no such ODE exists.

The first ones I have been able to figure out by reverse-engineering the solving process...

e.g. $\displaystyle e^t+exp(2t) \longrightarrow y''-3y'+2y=0$

However, I am having trouble with these two:

- $\displaystyle t^2 e^t$
- $\displaystyle exp(x^2)$

Any insight would be greatly appreciated!