Why can't all differential equations be solved perturbatively?

Question

I came across this question as a nice little worked example on how to solve a differential equation using perturbation methods.

My question is can we solve all ODE's in this way? When is it inappropriate to use such an approach?

Answer 1

As explained in an answer to your linked question, the first step is expansion in powers of $\epsilon$ (i.e., a power series expansion). What happens when the radius of convergence of that power series is too small to reach the differential equation you are actually trying to solve?

This is why it is called a perturbation method : you can solve in a neighborhood of a simple equation. But this technique gives no promises about making large deflections in the equation to solve.

I don't have an example handy (except for various hairy equations in renormalization). I imagine there is an example of finite radius of convergence near $$ y' = \frac{1}{1- \epsilon y} $$ since the right-hand side is $1 + \epsilon y + (\epsilon y)^2 + \cdots$ for $|\epsilon y| < 1$. (But I haven't checked and this example may be better behaved than I expect.)