- #1
Shackleford
- 1,656
- 2
Homework Statement
(1+x2) y'' + 2xy' = 0 in powers of x
Homework Equations
[itex] y'' = \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} [/itex]
[itex] y' = \sum_{n=1}^{\infty} na_nx^{n-1} [/itex]
The Attempt at a Solution
(1+x2) y'' + 2xy' =
[itex] (1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=1}^{\infty} na_nx^{n-1} = 0 [/itex]
[itex] (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 [/itex]
[itex] 2a_2 + 6a_3x + 2a_1x + \sum_{n=2}^{\infty} [(m+1)(m+2)a_mx + (m-1)ma_m + 2ma_m] x^{m} [/itex]
[itex]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 \\
[/itex]
[itex] a_{2n} = 0, a_{2n+1} = (-1)^n\frac{a_1}{2n+1} [/itex]