Section 6.1 The Method
The method is simple enough to remember. Say we want to solve the ODE
\begin{gather*}
P(x)y'' + Q(x)y' + R(x)y = 0,
\end{gather*}
around the ordinary point \(x = x_0\text{.}\) Then, to solve the ODE, substitute in
\begin{gather*}
y(x) = \sum_{n=0}^\infty a_n (x-x_0)^n.
\end{gather*}
After you substitute this form in, condense all the terms and find a “recurrence relation” for \(a_n\text{.}\) We'll see an example of this in the examples.
Be wary that these methods are often algebraically tedious.