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?