Question

Power series method to solve differential equations

Answer 1

Consider the second-order linear differential equation

{\displaystyle a_{2}(z)f''(z)+a_{1}(z)f'(z)+a_{0}(z)f(z)=0.\;\!}a_2(z)f''(z)+a_1(z)f'(z)+a_0(z)f(z)=0.\;\!

Suppose a2 is nonzero for all z. Then we can divide throughout to obtain

{\displaystyle f''+{a_{1}(z) \over a_{2}(z)}f'+{a_{0}(z) \over a_{2}(z)}f=0.}f''+{a_1(z)\over a_2(z)}f'+{a_0(z)\over a_2(z)}f=0.

Suppose further that a1/a2 and a0/a2 are analytic functions.

The power series method calls for the construction of a power series solution

{\displaystyle f=\sum _{k=0}^{\infty }A_{k}z^{k}.}f=\sum_{k=0}^\infty A_kz^k.

If a2 is zero for some z, then the Frobenius method, a variation on this method, is suited to deal with so called singular points.