method of Frobenius

If x_0 is an ordinary point of the ordinary differential equation, expand y in a Taylor series about x_0. Commonly, the expansion point can be taken as x_0 = 0, resulting in the Maclaurin series y = sum_(n = 0)^∞ a_n x^n. Plug y back into the ODE and group the coefficients by power. Now, obtain a recurrence relation for the nth term, and write the series expansion in terms of the a_ns.

Fuchs's theorem | ordinary differential equation