I have hard time trying to find the leading behavior of the following ODE as $x\rightarrow \infty$: $$ x^{7} y'' = e^{x} y$$ I've tried to use the substitution: $$ y = e^{S(x)}$$ and using the method of dominant balance I assumed: $$ (S')^{2} \gg S''$$ As a result: $$ (S')^{2} \sim \frac{e^{x}}{x^{7}}$$ Usually I would try to integrate but it seems like I'm missing something.
I saw a similar question in Advanced Mathematical Methods by Bender in page 140 but he didn't mention how to deal with it.
So I answered my question: We are looking at the limit $x\rightarrow 0$ and therefore $$\frac{e^{x}}{x^7} \sim \frac{1}{x^7}, \text{ as } x\rightarrow 0$$ and from here it is solvable.