Paper 2, Section II, $6 \mathrm{C}$

Find all power series solutions of the form $y=\sum_{n=0}^{\infty} a_{n} x^{n}$ to the equation

$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\lambda^{2} y=0$

for $\lambda$ a real constant. [It is sufficient to give a recurrence relationship between coefficients.]

Impose the condition $y^{\prime}(0)=0$ and determine those values of $\lambda$ for which your power series gives polynomial solutions (i.e., $a_{n}=0$ for $n$ sufficiently large). Give the values of $\lambda$ for which the corresponding polynomials have degree less than 6 , and compute these polynomials. Hence, or otherwise, find a polynomial solution of

$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+y=8 x^{4}-3$

satisfying $y^{\prime}(0)=0$.

*Typos? Please submit corrections to this page on GitHub.*