Lagrange identity for a Sturm-Liouville operator ($uL(v)-vL(u)=[p(u{v}^{\prime }-v{u}^{\prime }){]}^{\prime }$)

For $L(v)=(p{v}^{\prime }{)}^{\prime }+av$, direct differentiation gives $uL(v)-vL(u)=[p(u{v}^{\prime }-v{u}^{\prime }){]}^{\prime }$. The potential terms cancel. Integrating with common homogeneous Dirichlet boundary conditions eliminates the endpoint term. If $L(u)=0$ and $L(v)=f$, this gives $\int uf=0$, a necessary solvability condition for the inhomogeneous equation.

For the real-coefficient differential operator $Mv=(p{v}^{\prime }{)}^{\prime }+qv$, integration by parts givesIf $p$ tends to zero at both endpoints and $u,v,u^{\prime },v^{\prime }$ remain bounded there, the boundary term vanishes without prescribing endpoint values. Thus regular endpoint behaviour can supply the domain condition that makes the expression symmetric. Endpoint degeneracy alone is insufficient for arbitrary singular functions: for $p=x(1-x)$ and $v=\log x$, one has $p{v}^{\prime }=1-x\to 1$ at zero. A self-adjoint differential operator requires a specified admissible domain, rather than only a formal differential expression.