In literature I am using, to solve the following differential equation:
$$\frac{d^2 u}{d \rho^2} = \bigg(1 -\frac{\rho_0}{\rho} + \frac{l(l+1)}{\rho^2} \bigg)u$$ where $l$ is a constant integer, we examine the asymptotic form of the solutions as $\rho \to \infty$ and $\rho \to 0$. As $\rho \to \infty$ the constant term in brackets dominates, so (approximately) $$\frac{d^2 u}{d \rho^2} = u~~~ \text{which has general solution}~~u(p) = Ae^{- \rho} + Be^{\rho}$$ for large $\rho$. We want a normalizable solution, so we take $B=0$. Evidently, $$u(\rho) \approx Ae^{- \rho} $$ for large $\rho$. On the other hand, as $\rho \to 0$ the third term dominates so then we have the approximation: $$\frac{d^2 u}{d \rho^2} = \frac{l(l+1)}{\rho^2}u.$$ The general solution is $$u(\rho) = C \rho^{l+1} + D \rho^{- l}$$ again we take $D =0$, thus $$u(\rho) \approx C \rho^{l+1}$$ for small $\rho$. The next step is said to "peel off the asymptotic behaviour" by introducing a new function $v(\rho)$: $$u(\rho) = \rho^{l+1}e^{- \rho}v(\rho).$$ Is this a standard method of solving differential equations and what is the advantage of writing $u$ as a new function $v$? Is the the last question valid because we can account for the limit behaviour of $u$ by requiring $v(\rho) \to A \rho^{-(l+1)}$ for large $\rho$ and $ v(\rho) \to Ce^{\rho}$ for small $\rho$?
Thanks for assistance.
One cannot say that this is "a standard method of solving differential equations". This is one way among several. When facing a difficult differential equation, it is of use to try several approches until finding a successful one. Of course, the search could be unsuccessful and numerical calculus is often required.
For example, it is a common use to change of function $y(x)=x^ae^{bx}g(x)$ and determine the parameters $a$ and $b$ which leads to an ODE simpler than the initial one. This is often valuable for ODE's of generalized Bessel kind or more generally of hypergeometric kind. Of course, this isn't successful in many other cases of ODEs.
In the present case, considering the limit solutions at $0$ and at $\infty$ is advantageous because it avoid boring calculus to determine the above parameters $a=-(l+1)$ and $b=-1$. Nevertheless the transformed ODE with unknown function $v(\rho)$ remains complicated. It is much simpler to directly refer to a known form of ODE which solutions are some known special functions : The Coulomb wave functions or the Whittaker functions in the present case.
http://mathworld.wolfram.com/CoulombWaveFunction.html
http://mathworld.wolfram.com/WhittakerDifferentialEquation.html