1. (5 points) Find the extremal of the functional
2. (5 points) Consider the functional
• (2 points) Find the Euler-Lagrange equation of the above functional.
• (1 point) Find r such that y(x) = xr solves your Euler-Lagrange equation.
• (2 points) Find the extremal of J[y] satisfying the boundary conditions y(1) = 0,
.
3. (5 points) Determine the extremal of the functional
where α,β are nonzero constants for each of the following boundary conditions:
(a) y(0) = 0, y(1) = 1.
(b) y(0) = 0, y(1) is free.
(c) y(0) and y(1) are both free.
