I need to find the eigenvalue/eigenfunction pairs of the following DE:
$$x^2y''+3xy'+\lambda y=0, \text{ BC: }y(1)=0, y(e)=0 $$
Through my calculations I have ended up with $\lambda_n = n^2\pi^2 +1$ and the corresponding eigenfunctions as $y_n(x)=sin(n\pi ln(x))$ but when I tried the case when $n=1$ the pair didnt satisfy the equation, any tips on how to go about this?