Ordinary-derivative Robin data in a weighted energy form ($a(u,v)={\int }_b^{c}p(x)u^{\prime }{v}^{\prime }+p(c)\kappa u(c)v(c)$)

For $-(p{u}^{\prime }{)}^{\prime }=f$, $u(b)=0$ and $u^{\prime }(c)+\kappa u(c)=0$, integration by parts gives the displayed endpoint coefficient. Ordinary derivative data and conormal data $p(c)u^{\prime }(c)+\kappa u(c)=0$ are different boundary conditions. If $p$ is bounded above and below by positive constants and $\kappa \ge 0$, the Poincaré inequality on functions vanishing at $b$ proves continuity and coercivity. The symmetric form defines an energy norm, and Galerkin orthogonality gives exact best approximation in that norm.