My question is more or less similar to this one, which doesn't appear to have been answered beyond some comments about non-uniqueness. Let me try to be a bit more explicit though because I think perhaps my case has additional special properties that should make it fairly straightforward to solve.
So I am trying choose $y(x)$ to maximize:
$$ \int_{x_0}^{x_1} F(x, y, y')$$
Where $y'$ is the derivative of $y$, and everything is single-valued. In my case, $F$ has a particular form. There exist some functions $U, K$ such that
$$F(x, y, y') = U(x, y(x)) - K(x)[\frac{dU(x, y(x))}{dx}]$$
Apologies if my notation is poor. $\frac{dU}{dx}$ is equal to the total derivative of $U$ with respect to $x$ and is a function of $x, y, y'$.
The reason this is important is that it's relatively easy to show that this implies that solutions to $U_2 = 0$ (where $U_2$ is the partial derivative of U in its second argument) will solve the Euler-Lagrange equation. It turns out the $U_2 = 0$ is very easy to solve in my case, and the solution is unique. So all that remains to show is sufficiency.
Unfortunately the form above also implies that $F$ is linear in $y'$, which means the Legendre condition is a no-go. I tried looking for other conditions and generalized Legendre-Clebsch conditions, but have struck out on anything that can help me prove sufficiency.
Assistance in solving this particular form of the problem is fine, but really I would just love a relatively thorough reference on the issue, as similar problems arise in my research not infrequently. It seems there was a somewhat active literature in this area in the 60s and 70s (Jacobson, Kelley, Goh seem to come up a lot), but I'm having a hard time finding a great summary of how to get sufficiency in "poorly behaved" cases. I suspect that for my case in particular I may be able to argue from first principles instead of using these derived conditions, but don't have a great sense of how to make such arguments.
My sense is that the sufficient condition for a maximum in this case should boil down to something like $1-K'(x) > 0$ (I have in mind a discrete analog where this is the key property). Thanks, and sorry in advance for whatever norms and rules of posting, notation, etc. I am breaking, dumb or naive assumptions I'm making, lack of clarity, etc. Did my best.
Edit: Just wanted to add, I am also familiar with and own the textbook from Gelfand and Fomin, so a different reference would be ideal. But if someone can help, e.g. argue from the principles in that text (e.g. proving sufficiency directly via 'strong positivity'), that would be much appreciated as well.