# Problem with finding second solution to ODE

1. The problem is to find the series solution to the following differential equation
$$x^2 \frac{d^2 x}{dx^2}+x\frac{dy}{dx}+(x^2 - 1)y$$

3. Using the ansatz $$y = \sum _{\lambda = 0}^{\infty}a_{\lambda}x^{k+\lambda}$$ the
solution to the indicial equation was found to be k=1 and k=-1. I obtained a solution for k = 1, however I am having a problem with obtaining the second solution. The recurrence relation for k = -1 is $$a_{j}=-\frac{a_{j-1}}{(j-1)(j-2)+j-2}$$ which diverges for j = 2. In this case, is it possible to obtain a solution for k = -1?

## Answers and Replies

vela
Staff Emeritus
Science Advisor
Homework Helper
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If the roots of the indicial equation differ by an integer, the second root may not yield a solution, which appears to be the case here. Your textbook probably covers how to get a second solution in such a case, and it's mentioned on this Wikipedia page: http://en.wikipedia.org/wiki/Frobenius_method#Double_roots

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