 Solution

Let $D=d/dx$. Repeated integration by parts gives the formal adjoint rulesThereforeEquality of the coefficients of $D$ in $L$ and $L^{\ast }$ requiresso $q=p^{\prime }$. This also makes the zeroth-order coefficients equal because $p^{\prime \prime }-q^{\prime }=0$. Hence, under boundary conditions that remove the boundary terms,This is the self-adjoint fourth-order scalar differential operator $D^4+p{D}^2+p^{\prime }D+r$.