Mike McNulty, who is a postdoc working with me, showed me the following trick for looking at asymptotic behavior of ODEs near singular points that he found; my question: does it have a well-known name in the literature?

Preliminary

Consider an ODE $u'' = Vu$. The method of reduction of order / quadrature tells us that, given one solution $u_0$ that does not vanish on an interval $(a,b)$, one can generate another linearly independent solution $v$ by solving $$ \left( \frac{v}{u_0} \right)' = \frac{1}{u_0^2} \tag{*}$$

Method

Suppose now that $V$ is possibly singular, so that on the interval $(0,1)$ the solution $u_0$ looks like $u_0(x) = x^\lambda \omega(x)$, where $\omega$ is bounded away from zero, and $\lambda > 0$. Then we can write-down $v$ using the following Laurent-series like expansion (basically by fiddling around with (*) and differentiating by parts). Here $M$ is an arbitrary positive integer denoting the order of expansion:

$$ \left(\frac{v}{u_0} \right)' = \left[\sum_{k = 1}^{M} (-1)^{k-1} \frac{x^{k - 2\lambda}}{(k-2\lambda)(k-1-2\lambda) \cdots (1-2\lambda)} \left( \frac{1}{\omega^2(x)}\right)^{(k-1)}\right]' + (-1)^M \frac{x^{M-2\lambda}}{(M-2\lambda)\cdots(1-2\lambda)} \left(\frac{1}{\omega^2(x)}\right)^{(M)} $$

Note

This feels like something that should be classically known, but I can't find anything like it in the three ODE texts I own (Hartman, Birkhoff-Rota, Hale; though possibly I was looking at the wrong parts). Any references would be appreciated. I feel that it is a lot like the method of Frobenius, except we do not assume analyticity.