As stated above, can I solve things like $p(x) = e^{h(x)}$ without approximation? If not is there an explanation?
Lets say here that $h$ and $p$ are polynomials.
I can't say I've done much more than try the typical algebraic tricks on easy equations, but nothing normal worked. I also determined by graphing some basic equations that often solutions do not exist, so maybe there is something to be said there about when solutions exist.
So I am interested if there is some trick method here that works, either in specific cases, or in general. Or any theory behind these types of equations involving analytic functions with infinite series expansions such as $exp$.