(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?

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# Homework Help: Power Series Solution

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