A series solution of y' = xy

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Question from sasha, a student:

Hi,
I need help with the following question.
I've to find the power series solution of the differential equation: y' = xy.
I don't know how to find the recursive equation. Can you please help me. Thanks

Sasha,

Suppose the solution is series then series and series . Hence if y' = xy then

For the series on the right side change the index to k and rewrite it as series

The left side can be written series Let k = n - 2 and this becomes series Hence y' = xy can be written

Now you can see the coefficients that must be equal

a_k = (k + 2)a_k+2 for k = 0, 1, 2 ···

I hope this helps,
Harley

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