The characteristic equation in the calculus of variations (at least that's what my book calls it) is $$\frac {\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'}=0$$ where $F=F(x,y,y')$ in general.
My book says that in the case that $F$ does not explicitly depend on $x$ that this simplifies to $$F-\frac {dy}{dx}\frac{\partial F}{\partial y'}=c$$
Clearly they've integrated here (or else I'm not understand how the constant $c$ was obtained), but how do you integrate $\int \frac{d}{dx}\frac{\partial F}{\partial y'}dy$?