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