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**1. The problem is to find the series solution to the following differential equation**

$$ x^2 \frac{d^2 x}{dx^2}+x\frac{dy}{dx}+(x^2 - 1)y $$

$$ x^2 \frac{d^2 x}{dx^2}+x\frac{dy}{dx}+(x^2 - 1)y $$

**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?

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?