Asymptotic Analysis of 2nd Order Differential Equation with Irregular Singular Point

Question

I am asked to find the leading behavior of the solution $y(x)$ to this differential equation that is asymptotically the largest as $x \to 0$: $$x^2\frac{d^2y}{dx^2} + \frac{dy}{dx} + \frac{3}{16}y = 0$$ From assuming that $y(x)$ has the form $e^{S(x)}$, taking derivatives, and substituting them back into the original equation, I get the following equation: $$x^2S''(x)e^{S(x)} + x^2e^{S(x)}(S'(x))^2 + S'(x)e^{S(x)} + \frac{3}{16}e^{S(x)} = 0$$ I then made the assumption that $S''(x) \ll (S'(x))^2$, which gives this asymptotic relationship: $$(S'(x))^2 \sim \frac{-S'(x)}{x^2} - \frac{3}{16x^2}$$ I'm not sure where to go from here. Most examples I can find online don't have a factor of $S'(x)$ on the RHS, so taking the square root and integrating becomes simple. I also want to find a correction factor for the solution in series form. Some guidance on this would be appreciated.

Answer 1

This DE can actually be solved in a closed form involving Bessel functions: according to Maple, the general solution is

$$ \frac{c_{1} {\mathrm e}^{\frac{1}{2 x}} \left(\left(x -2\right) I_{\frac{1}{4}}\! \left(-\frac{1}{2 x}\right)-2 I_{-\frac{3}{4}}\! \left(-\frac{1}{2 x}\right)\right)}{\sqrt{x}}+\frac{c_{2} {\mathrm e}^{\frac{1}{2 x}} \left(\left(x -2\right) K_{\frac{1}{4}}\! \left(-\frac{1}{2 x}\right)+2 K_{\frac{3}{4}}\! \left(-\frac{1}{2 x}\right)\right)}{\sqrt{x}}$$

As $x \to 0+$, the coefficient of $c_1$ is asymptotic to: $$ - \frac{4\; i}{\sqrt{\pi}} + \frac{3\; i}{4 \sqrt{\pi}} x + \ldots $$

while the coefficient of $c_2$ is asymptotic to $$ -\frac{3\; i}{4} \sqrt{\pi} x^2 e^{1/x} - \frac{105\; i}{64} \sqrt{\pi} x^3 e^{1/x} + \ldots$$