I am having trouble with F = y - y'
If I set F = |y - y'| and
do $\int F(y,y') dx$,
it seems obvious that $y=\exp(x)$ should be extremal
since $F=|\exp(x)-\frac{d \exp(x)}{dx}| = |\exp(x)-\exp(x)|=0$
But I am not sure how to apply Euler-Lagrange Equation...
Is this any way to solve generalization of this problem?