Legendre Eigenvalue problem

Question

I have the eigenvalue problem, $\frac{d}{dx}\big((1-x^2)\frac{du}{dx}\big)+\lambda u=0$, on $[-1,1]$ subject to single boundary condiction $u(-1) = u(1)$. Assume that there is an eigenfunction of the form $u(x)=a_0+a_1x+a_2x^2$. Find the possible eigenvalues for such an eigenfunction. To solve this problem, I plug the $u(x)$ and the 2nd derivative $u(x)$ into the $\frac{d}{dx}\big((1-x^2)\frac{du}{dx}\big)+\lambda u=0$, solve for $x$. But I couldn't get any eigenvalues. Any help will be appreciated.

Answer 1

Well, I assume it is sufficiently late I am not doing your homework for you. The boundary condition tells you that $a_1=0$, so it's missing from your Ansatz. Plugging the surviving piece into the equation yields $$ \partial ((1-x^2)2a_2)=-\lambda(a_0+a_2x^2), $$ so, comparing powers, $\lambda=6$ and $a_2=-3a_0$.

In fact, do you now see how your particular case (n=2) falls into the general order-n polynomial solution $$ P_n\propto \partial^n (x^2-1)^n $$ of $$ P_n(x)= \partial ((1-x^2)\partial P_n) +n(n+1)P_n=0 ~~? $$