Calculus of variations Euler Lagrange equation How can I find an extremal for the following functional:
$$J[y] = \int_0^1 e^y (y')^2 dx $$
with the boundary conditions $y(0)=1, y(1)=\log(4)$?
I have tried computing the Euler-Lagrange equation but it becomes overly complicated. 

The E-L equation is $- e^y (y')^2 - 2 e^y y'' = 0$.  Divide by $e^y$ and first solve the first-order equation for $y'$, then integrate to get $y$.  Then put in the boundary conditions.
