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Consider the equation: $$(1-x^2)y'' -xy' +\alpha^2 y=0$$ where $\alpha$ is a real number.

For $|x|<1$ and all values of $\alpha$, look for a fundamental set of solutions $y_1$, $y_2$ which are power series in $x$. Prove that they are linearly independent. You can leave the formula for the coefficients in recursive form, but compute the first three terms of $y_1$, $y_2$ explicitly in terms of $\alpha$.

What does it mean when the question asserts to look for a fundamental set of solutions that are a power series in $x$? Likewise, what does the question mean when it states to leave the coefficients in recursive form?

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It means find a solution of the form $$ y(x)=\sum_{n=0}^\infty a_nx^n, $$ were you are supposed to find the coefficients $a_n$. To leave them in recursive form means that you do not need to find them explicitly, but to obtain a recurrence formula where $a_n$ is given in terms of $a_0,\dots,a_{n-1}$.

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  • $\begingroup$ Proving linear independence is easy for me. I will simply take the Wronskian and show that it does not equal to zero. But how would I go about finding y1,y2,y3. Shouldn't there at most be only two solutions? I'm lost at making my first step $\endgroup$ – seekingalpha23 Aug 5 '18 at 14:04

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