This question comes from Bender and Orszag's Asymptotic Methods and Perturbation Theory. I'm practicing applying the method of dominant balance to study behavior as $x\to \infty$ for systems which have an irregular singular point at infinity. My question is about how to find what Bender and Orszag call the leading behavior, after having found the controlling factor, especially in systems with high order derivatives. For example, the system $x^3 d^5y/dx^5 = y$ has an irregular singular point at $\infty.$ We ansatz $y=e^{S(x)}$, and make the standard assumption that $S'' \ll (S^\prime)^2$ as $x\to \infty$.
It is easy to compute that $y'' = (S''+(S')^2)e^s \sim (S')^2 e^s$, which then implies that $y''' \sim ((S')^3+2S'S'')e^S$. However, by assumption, $S'S''\ll (S')^3$, so $y''' \sim (S')^3 e^S.$ Continuing in this way, we find that $d^5y/dx^5 \sim (S^\prime)^5 e^S$ as $x\to \infty$. The asymptotic relation for $y$ then reads $x^3 (S')^5 e^S \sim e^S$. This is easily solved for $S$: $$S \sim \frac{5 \omega x^{2/5}}{2},$$ where $\omega$ is a $5$th root of unity.
Now that we have the controlling factor $S\sim\frac{5 \omega x^{2/5}}{2}$, we seek the leading behavior by searching for a $C(x)$ such that $S(x) = \frac{5 \omega x^{2/5}}{2} + C(x)$ and $C(x)\ll \frac{5 \omega x^{2/5}}{2}$ as $x\to \infty$.
This is where I have a question. In computing the aymptotic relation satisfied by $S$, we were able to simplify the derivatives of $e^S$ by repeatedly making use of the assumption that $S''\ll (S')^2$. However, now that I want to find $C$, it seems that I can't do the same. Indeed, if we do, then we would find ourselves with the relation $$(S')^5 \sim x^{-3}$$ $$((\frac{5 \omega x^{2/5}}{2} + C)')^5 \sim x^{-3}$$ which after some simplification leaves us with $$C' \sim 0,$$ which is obviously useless. It therefore seems like what we need to do is actually fully compute $\frac{d^5}{dx^5}\left ( \exp(\frac{5 \omega x^{2/5}}{2} + C(x)) \right )$ and then do dominant balance on the result. This seems like a huge pain, because the derivative will generate a heap of terms. It feels like this can't be how the method works.
My question is: What is the "right" way of finding an asymptotic relation for $C(x)$ in situations like this? Thank you for your time.