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I'm trying to get the general solution to the differential equation, $${\partial^2y(x) \over \partial x^2}+4x{\partial y(x) \over \partial x} + y(x) = 0$$ using the power series method, about the point $x = 0$, and I am to give at least the first four terms in each series.

Here is my attempt at the problem, however I feel like I'm going about this in the wrong way. What do I need to do to solve this problem?

$$ \sum_{n=2}^\infty n(n-1)a_nx^{n-2} + 4x\sum_{n=1}^\infty na_nx^{n-1} + \sum_{n=0}^\infty a_nx^{n} =0 $$ We have the following equality, below ... $$4x\sum_{n=1}^\infty na_nx^{n-1} =\sum_{n=1}^\infty 4na_nx^{n} =\sum_{n=0}^\infty 4na_nx^{n} $$ We take this and put it in the original equation, while also getting the summing indexes to be the same i.e. n = 0 to inf. $$ \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^{n} + \sum_{n=0}^\infty 4na_nx^{n} + \sum_{n=0}^\infty a_nx^{n} = 0 $$

Gathering like terms, we find that ... $$a_{n+2} = -{(4n+1) \over (n+2)(n+1)}a_n$$

I'm not really sure where to go from here. I've plugged in various n values (n = 1 to 9), but I don't really see any overall pattern except for the denominator being $(n+2)!$. Where do I need to go from here?

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  • $\begingroup$ There are general formulae but, as you can expect, they involve the Gamma function (with some non integer arguments). Are you supposed to use them ? By the way, are there initial conditions for the ODE ? $\endgroup$ – Claude Leibovici Dec 4 '15 at 7:14
  • $\begingroup$ I've never even heard of the "Gamma function", so I'm guessing I'm not suppose to use it. The only initial condition I'm given is "find the solution about the point x = 0". $\endgroup$ – Collaptic Dec 4 '15 at 7:22
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    $\begingroup$ So, just stop at this point. You have a nice recurrence relation. $\endgroup$ – Claude Leibovici Dec 4 '15 at 7:24

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