Answer Happy

(a) Let T> 0. A model in economics seeks to optimise investment M(t) by calculating a stationary path of the functional S[M] over the fixed time interval 0 ≤ t ≤ T where (b) T S[M] = [₁ dt a(bM — M' — c)², M(0) = Mo, and where a, b, c, Mo are positive constants and M' = dM/dt. (i) Calculate the stationary path M* of S[M]. (ii) Calculate S[M₂] and M*[T]. (iii) Show that M、 gives global minimum of the functional S[M]. Let a 0. Consider the functional S[y] = Ⓡ dx xy2², y(a) = 0, subject to the constraint a C[y] =d dx xy² = 1. (i) By using the Rayleigh-Ritz method with trial function z = z(x; A, ß) = A(a²-x²)³, where A > 0 and 3 > 1/2, find an approximation to the smallest eigenvalue of the associated Sturm-Liouville system. (ii) By an appropriate choice of ß, find the closest approximate value of the eigenvalue from this family of trial functions.