Solution for a general variational equation of 2nd order

[brunotolentin.4]

In a book of math, I found a kind very very crazy of equation, an "variational equation of second order"

So, my question is: exist solution for an general equation like this:

$$A \frac{d^2}{dx^2} \frac{\partial F}{\partial y^{(2)}} + B \frac{d^1}{dx^1} \frac{\partial F}{\partial y^{(1)}} + C \frac{d^0}{dx^0} \frac{\partial F}{\partial y^{(0)}}= 0$$
(with A, B and C being constants)

EDIT: my other question is: what you know about this kind of equation? I never saw nothing like this before...

 

[Orodruin]

This completely depends on the function $F$. If you want to know more about this type of equations, you need to study a textbook or similar material regarding variational calculus.