Math292 Textbook The Method

Section 6.1 The Method

The method is simple enough to remember. Say we want to solve the ODE

\begin{matrix} P (x) y^{″} + Q (x) y^{'} + R (x) y = 0, \end{matrix}

around the ordinary point

x = x_{0} .

Then, to solve the ODE, substitute in

\begin{matrix} y (x) = \sum_{n = 0}^{\infty} a_{n} (x - x_{0})^{n} . \end{matrix}

After you substitute this form in, condense all the terms and find a “recurrence relation” for

a_{n} .

We'll see an example of this in the examples.

Be wary that these methods are often algebraically tedious.