Question

I am to find the series for y''+2xy=0

Answer 1

use the guess \[y = \sum_{n=0}^{\infty} C_nx^n\]

Answer 2

find the first and second derivative for \(y\) which is \[y'' = \sum_{n=2}^{\infty}n(n-1)C_nx^{n-2}\]

Answer 3

substitute into your given differential equation to get: \[\sum_{n=2}^{\infty}n(n-1)C_nx^{n-2}+2x \sum_{n=0}^{\infty}C_nx^n = 0\] distribute the \(2x\) inside the sum
\[\sum_{n=2}^{\infty}n(n-1)C_nx^{n-2}+2\sum_{n=0}^{\infty}C_nx^{n+1}=0\]

Answer 4

Are you with me so far?