Given the boundary value problem $$(1+x)\frac{d^{2}u}{dx^{2}}+\frac{du}{dx}+\lambda u=0, \quad u(0)=u(1)=0 $$ where $\lambda\in\mathbb{C}$, on the interval $[0,1]$. Then the BVP has
- A non-trivial solution for all $\lambda\in\mathbb{C}$
- Only a trivial solution for all $\lambda\in\mathbb{C}$
- A non-trivial solution for only finitely many values of $\lambda$
- A non-trivial solution for countably many values of $\lambda$
I know how to solve the equation when it is in the form
$$\frac{d^{2}u}{dx^{2}}+\lambda u=0 $$
but don't know how to solve this problem, one thing I know about the equation is that it can be written in the form
$$\left[(1+x)\frac{du}{dx}\right]'+\lambda u = 0$$
and here in this equation we have $p(x)=(1+x)$ and $q(x)=0$. Next I don't know how to solve further. Please help.