Find the extremal of the Functional $J[y(x)]=\int_{0}^{1}(2e^{y}-y^{2})dx$ subject to $y(0)=1$ and $y(1)=e$.
Using Euler's equation we get the answer $y=e^{y}$.
My question is how to check the boundary condition? Is this actually a solution to the original problem? In general how to tackle a problem in which the answer obtained through Euler's equation does not contain x?