Chapter 11.4, Problem 10E

In Problems 7-10, find the adjoint operator and its domain for the given linear differential operator L whose domain consists of functions that have continuous second derivatives and satisfy the indicated boundary conditions.

$\begin{array}{l}L\left[y\right]\left(x\right)=x^2{y}^{″}\left(x\right)+2x{y}^{\prime }\left(x\right)+\frac{5}{4}y\left(x\right);\\ y\left(1\right)=0,y\left(e^{\pi }\right)=0\end{array}$