(adsbygoogle = window.adsbygoogle || []).push({}); 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:

[tex]y' - 4 x y - 2 = 0[/tex]

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

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

Or in other words...

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

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Power Series Solution

**Physics Forums | Science Articles, Homework Help, Discussion**