physicist here.
I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come from an asymptotic analysis: For example if I try to analize then a equation like $\frac{dx}{dt}+x=\frac{t}{x}$, I would start making $x \rightarrow 0$, and then solving the equation $\frac{dx}{dt}=\frac{t}{x}$. After that, I'll take dominant term $u(t)$ in the solution to make a variable change $x(t)=u(t)f(t)$ and then simplify (and solve) the equation.
The problem that now I'm facing is the following differential equation, at $ x \rightarrow 1$:
$$ x^2 s \frac{d^2u(x)}{dx^2} + xs\frac{du(x)}{dx}+\frac{u(x)x}{c} \left ( \frac{p}{1-x} + \frac{q}{(1-x)^2}\right )=-K u(x)$$
My try: First I tried to change all that $x$ by 1 except the denominators, then solving the differential equation. It was completely impossible anyway. I've thought on power series, but that $1-x$ are very annoying. I've finally tried with Mathematica. I discovered that substituting the $x$ by 1 is NOT a good idea. Also now I'm pretty sure that the solution is an hypergeometric. However I could not solve the entire equation, I had to eliminate the $p/(1-x)$ term (the computer could not simplify in that case).
I need a good way to attack this. Any ideas?