(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the general solution near x = 0 of y'' - xy' + 2y = 0 (using power series).

Answer:

y = a_0 * y_1(x) + a_1 * y_2(x)

where

y_1(x) = 1 - x^2 and y_2(x) = x - 1/6 * x^3 - 1/120 x^5 - 1/1680 * x^7 - ...

2. Relevant equations

Power series. Sigma notation for summations. Polynomial derivatives.

3. The attempt at a solution

My attempt is attached. I get y(x) = ke^x and I think I'm wrong but I'm not sure since the answer my book gives doesn't use sigma notation.

I'd appreciate if someone could tell me if I'm right or wrong and if I am wrong, where I went wrong.

Thanks in advance!

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# Finding general solution to a differential equation using power series

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