Consider the following question
I have worked out that the Eigenvalues for this DE are $\lambda_n = 4 - \frac{n^2 \pi^2}{4}$ and that the corresponding Eigenfunctions are $y_n = B \cdot Sin(n \pi)$ for some constant $B \in \mathbb{R}$.
I have also found the Sturm-Louville form of the equation, which is $$ \frac{d}{dx} \left[ e^{4x} y' \right] + e^{4x} \lambda y = 0$ $$ Assuming that my results above are correct, how am I to attempt the next part of this question?
Perform the integral $$\int_{0}^{2}-e^{-4x}\sin\left(\lambda_nx\right)\sin\left(\lambda_mx\right)dx$$ and see what pops out.