(a) Consider the functional S[y] = [* da (y² +9y² + 2(3x²+x+ dx 10 -)) 3 11 with boundary conditions y(2) = 1, y(4) = 9

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A Consider The Functional S Y Da Y 9y 2 3x X Dx 10 3 11 With Boundary Conditions Y 2 1 Y 4 9 1 (129.4 KiB) Viewed 24 times

(a) Consider the functional S[y] = [* da (y² +9y² + 2(3x²+x+ dx 10 -)) 3 11 with boundary conditions y(2) = 1, y(4) = 9 (i) Calculate the Gâteaux differential and hence obtain the Euler-Lagrange equation for a stationary path of the functional S[y], giving all steps in the procedure. (ii) Solve the Euler-Lagrange equation to obtain the stationary path. (b) Consider the functional S[y] = S dx (10 x y² – x³y¹²) with boundary conditions y(1) = 0, y(ln (6)) = sin (3 In (6)) 2 (i) Calculate the Jacobi equation. (You do not need to find or solve the Euler-Lagrange equation for this functional.) (ii) Find the solution of the Jacobi equation satisfying the initial conditions, and, if possible, use the Jacobi equation to classify the stationary path.