Consider the functional S[y] = ay(1)² + √² dx By, y(0) = 0, == with a natural boundary condition at x = 1 and subject to the constraint C[y] = √y(1)² + [* dx w(x) y² = 1, where a, ẞ and y are nonzero constants. Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y B +Aw(x) y = 0, y(0) = 0, (a-yλ) y(1) + ßy' (1) = 0, dx² where A is a Lagrange multiplier.

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