Find power series representations of the general solution

[Shackleford]

Homework Statement

(1+x²) y'' + 2xy' = 0 in powers of x

Homework Equations

$y'' = \sum_{n=2}^{\infty} (n-1)na_nx^{n-2}$

$y' = \sum_{n=1}^{\infty} na_nx^{n-1}$

The Attempt at a Solution

(1+x²) y'' + 2xy' =

$(1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=1}^{\infty} na_nx^{n-1} = 0$

$(1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=2}^{\infty} (n-1)na_nx^{n} + 2 \sum_{n=1}^{\infty} na_nx^{n} = 0$

$2a_2 + 6a_3x + 2a_1x + \sum_{n=2}^{\infty} [(m+1)(m+2)a_mx + (m-1)ma_m + 2ma_m] x^{m}$

$a_0 = a_0 \\ a_1 = a_1 \\ 6a_3 + 2a_1 = 0 \\ 12a_4 + 6a_2 = 0, a_4 = 0 \\ 20a_5 + 12a_3 = 0 \\$

$a_{2n} = 0, a_{2n+1} = (-1)^n\frac{a_1}{2n+1}$

 

[NascentOxygen]

Hi @Shackleford . We're all waiting for the other shoe to drop ...

... is there a question coming?