Solving ODE Near x=0: Series Solution

[BColl]

Homework Statement

Obtain solution valid near x=0

Homework Equations

(x²+1)y''+6xy'+6y=0

The Attempt at a Solution

y"+6x/(x²+1)y'+6x/(x²+1)=0
In representing the solution in series notation, I'm not sure how deal with the rational function because I know I need to have all of the x terms inside the summation and create powers of x that are all the same to create a single summation. The denominator of a rational function does not separate, so how would adjust the summation to take this into account?

 

[vela]

Substitute the series into the original equation, not the one with x²+1 in the denominators.

(I'm not sure exactly what your question is, but I took a stab at what I thought you meant.)