How do you find a Power Series solution of a linear differential equation?
1 Answer
To find a solution of a linear ordinary differential equation
around the point

If the point
#x_0# is an ordinary point for the differential equation, that is, all#a_i(x)# are analytic around#x_0# (their Taylor Series around#x_0# has a non zero convergrence radius), then we can use the ordinary power series method, described below. 
If the point
#x_0# is a regular singular point for the differential equation, that is,#x^i a_i(x)# are analytic around#x_0# , then we should use the Frobenius method (which will not be described in detail, due to it being more complicated). 
If the point
#x_0# is an irregular singular point, nothing can be said about the solutions of the differential equation.
For the ordinary power series method, start by assuming the solution of the differential equation to be of the form
Compute the
Applying the computed derivatives to the differential equation should give a recurrence relation for the coefficients
Solving that recurrence relation should give at least one solution for the differential equation that lies in the interval
This method is generally used for differential equations with polynomial coefficients (that is,
A simple example (generally solved by more elementary methods) to illustrate the recurence relations that appear for the coefficients
Finding the solution around
Computing the derivatives and applying them to the DE, we get:
Changing the index of the first sum using the relation
Since
Which is vallid if and only if
For the
Therefore:
wich is the well known solution for this problem.