I see the following regarding null Lagrangians in Courant and Hilbert's book, in the chapter on the Calculus of Variations:
The Euler differential expression for an integrand $F(x, y, y', …)$ may vanish identically for every admissible argument function. Since admissible argument functions can be constructed for which the quantities $y, y',\ldots$ take on prescribed values at an arbitrary point $x$, the identical vanishing of the Euler expression $[F]_y$ for all functions $y$ is equivalent to the identical vanishing of this expression if $x, y, y',\ldots$ are regarded as independent parameters. The same statement holds if the argument function depends on several independent variables. The simplest case is that of the integrand $F(x, y, y’)$. If the expression $F_y - F_{y’x} - F_{y’y}y’ - F_{y’y’}y’’$ vanishes, it follows that $F_{y’y’} = 0$ and therefore that $F$ is of the form $F = A(x, y) + y' B(x, y)$.
Although it apparently "follows" in some straightforward way, I'm having trouble seeing why $F_{y’y’} = 0$ should be a consequence of the vanishing of the variational derivative, i.e., of $F_y - F_{y’x} - F_{y’y}y’ - F_{y’y’} y’’\equiv 0$. Any insights would be appreciated.