TITLE: Which ODE Solution Method to Use?
QUESTION [5 upvotes]: It has been awhile since I've taken a course in differential equations, and I have problem, which requires I solve an ODE (after transforming a PDE) of the following structure:
$$
f(x) - (x + c_1)f'(x) = c_2xf''(x)
$$
where $c_1$ and $c_2$ are nonzero constants.
I'm not looking for the actual solution, I'm looking for the name of the solution method/procedure so that I can look into how to solve these types of ODEs.  Any pointers? 
Thanks!

REPLY [2 votes]: More commonly your equation would be written with the coefficient of the highest derivative normalized to 1, which makes it looks like
$$ f''(x) + \left( c_3 + \frac{c_4}{x}\right) f'(x) + \frac{c_5}x f(x) = 0 $$
with singular coefficients on the lower order terms. Hence they are sufficiently regular to be regarded as Fuchsian ODEs. So I would suggest you look into the literature for Fuchsian equations, and for "ordinary differential equations in complex domain".

Ah, it being second order and homogeneous helps also. It turns out the general class of "second order linear homogeneous ordinary differential equations with rational function coefficients that admits 'closed form solutions'" can be solved algorithmically.