Solving an ODE studying the asympotic behaviour I've seen that in some cases, one tries to study the behaviour of ODE at infinity and at zero, in order to find simplify the equation. For example:
$$y''=\left(1-\frac{a}{x}+\frac{b(b+1)}{x^2}\right)y$$
In the limit $x\to\infty$:
$$y''\approx y \implies y(x)=Ae^{x}+Be^{-x}$$
In the limit $x\to 0$:
$$y''\approx \frac{b(b+1)}{x^2}y \implies y(x)=Cx^{b+1}+Dx^{-b}$$
In this particular problem, we want $A=D=0$. So we try to find solutions of the type: $y(x)=x^{b+1}e^{-x}u(x)$
If we plug this in the original equation:
$$xu''+2(b+1-x)u'+(a-2(b+1))u=0$$
We have transformed our problem into the associated Laguerre equation. But how could we know the $u(x)$ ODE would have a solution? Is this method always true and why isn't it usually taught (I've only seen it in Quantum Mechanics)?
 A: Mathematically this corresponds to the analysis of singular points of linear ODEs in the extended complex plane of the independent variable. 
Loosely speaking, if for some particular linear ODE the number of such singularities turns out to be small and they are "not complicated" (which is the case in a number of quantum-mechanical problems), the equation will be solvable in terms of elementary or classical special functions. In such cases "subtracting the asymptotics" reduces the equation to one of the canonical forms with simple singularity structure - roughly Gauss hypergeometric equation or one of its degenerations. 
You can think of this procedure (maybe the right name for it would be the Riemann-Hilbert approach) as a kind of Liouville theorem-type argument: if you have a rational function on $\mathbb{P}^1$, it is fixed up to a constant by the negative parts of its Laurent expansions at poles. 
However, if the singularity structure of an ODE is complicated, the asymptotics is not helpful in constructing the solution explicitly. Not because we are not able to find the analogs of "negative parts" (this is precisely what the asymptotic analysis does), the problem is that we don't have available functions that "interpolate" between them.
