Consider the functional S[y] = ay(1)² + [[* dx ßy², y(0) = 0, with a natural boundary condition at x = = 1 and subject to the constraint C[y]=ry(1)² + [* dx w(x) y² = 1, where a, ẞ and y are nonzero constants. Euler-Lagrange equation β d²y dx2 +\w(x)y=0, y(0) = 0, (a− yλ) y(1) + ßy' (1) = 0, where is a Lagrange multiplier. Euler- Lagrange eft. is py" + 1 way=0 7 y" + dy , since A = 1, W(X) = 1 The general solution is y (x) = As in (√xx) + B cos (√xx) where A and B when y =0 B = 0 > are constants. heme, yox) = A sin (Dx) 110=0 When (α-rd) y (1) + By '10. (1-1) A sin √π + y'α> (1) = 0 since α = , x = 8 = p²l ⇒ (1-1) A sint + A√ cos t = 0 A Sin√√ C[y]=1 1 √ cos√λ = 0 1 - A² So Sin ² (√XX) = = ⇒ 1 = A² The nontrivial Az √2 stationary parth is g(x)= √2 sin (√xx)