Frobenius & Special Functions · Superchallenge problem

Consider the boundary value problem $x^2y'' + xy' + \lambda y = 0$ on $1 \le x \le e^{\pi}$ with $y(1) = 0$ and $y(e^{\pi}) = 0$. Let $\lambda_1 < \lambda_2 < \cdots$ be all the values of $\lambda$ for which a nonzero solution exists. Compute $\displaystyle\sum_{k\ge1}\frac{1}{\lambda_k}$.