Question

Find two power series solutions of the given differential equation about the ordinary point

Short answer

Answer

Therefore, the solution is:

Step-by-step solution

given information

The given equation is:

identify all the power series we'll need to solve the problem.

Substitute our power series for the original equation.

Find the first and second power series

First power series

Second power series

combine the two power series.

locate the first term of the first power series withas the index We integrate the two power series and extract a common factorafter discovering the term.

The two equations equal to.

Find a recurrence relation.

 find 

,and

To evaluateto

find 

,and

to

Therefore,