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# Second Order Differential Equation - Power Series Solution

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Hi there, I have a question regarding Differential Equations which can be located here http://none-questions.blogspot.com/2011/09/d-q7.html. can someone please take a look? Thanks.

##### Solution Summary

We solve a given second order ordinary differential equation by deriving and solving a recurrence in the power series of the solution.

##### Solution Preview

We wish to solve the following differential equation using power series:

xy'' - 2y + 9x^5 y = 0.

First write out y(x) as a power series, i.e.

y(x) = sum_{k=0}^infty(c_k x^k).

Then we have

y'(x) = sum_{k=1}^infty[k c_k x^(k-1)]
and

y''(x) = sum_{k=2}^infty[k(k-1) c_k x^(k-2)]

whence

0 = xy'' - 2y + 9x^5 y
= sum_{k=2}^infty[k(k-1) c_k x^(k-1)] - 2 ...

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