# Homework Help: Power Series Solution

1. Mar 1, 2012

### Char. Limit

1. The problem statement, all variables and given/known data
I am trying to find the power series solution to y' = 4 x y + 2, with the initial condition of y(0)=1.

2. Relevant equations

3. The attempt at a solution

Simple enough, I say, as I arrange the equation so I have 0 on one side. I get something like this:

$$y' - 4 x y - 2 = 0$$

I then assume that $y = \sum_{n=0}^\infty a_n x^n$. I also find that $y' = \sum_{n=0}^\infty (n+1) a_{n+1} x^n$ and I pick, for two, a series like $\sum_{n=0}^\infty \frac{1}{2^n}$. Subbing this all in, I get:

$$\sum_{n=0}^\infty \left(a_n - 4 \left(n+1\right) x a_{n+1} - \frac{1}{2^n}\right) x^n = 0$$

Or in other words...

$$\left(a_n - 4 (n+1) x a_{n+1} - \frac{1}{2^n}\right) = 0$$

But this doesn't look right. There's an "x" in there that shouldn't be there. What's the best way to remove the x?

2. Mar 1, 2012

### tiny-tim

Hi Char. Limit!

… you needed $xy = \sum_{n=0}^\infty a_n x^{n+1}$