Eigenvalue problem for the Euler equation $x^2y''+3xy'+\lambda y=0$ on $[1,e]$

Find the solutions of

\[ x ^ { 2 } y ^ { \prime \prime } + 3 x y ^ { \prime } + \lambda y = 0\tag{38} \]

with \(x \in [ 1 , e ]\) such that \(y ( 1 ) = y ( e ) = 0\)