Question 5. - 15 marks consump Dansi conditi Let a, b and w be constants, with w 0. (a) Show that the Euler-Lagrange equation for the functional is Stylda (y+y+2y(a sin wz + b sinh wz)), y(1) 0, y(2) = 1, = = y"-way asin wx+bsinhwa, y(1) = 0, y(2)=1. (b) Solve this equation to show that the stationary path is y = o sinh (@(2 - 1)) + B sinh (@(2 → 2)) - where [4] a sin wa 22 bar cosh wa + 2w [6] 1 a sin 2w b cosh 2w απ + sinha 2w2 لیا 1 bcosh w a sin w B= sinhư 2w 2w2 (c) Determine whether S[y] has a maximum or a minimum on this stationary path, and whether it is global or local. [5] Find the solution of the equations. 1) x'y " + 3xy+5y=8x 2) dy = ty-3 yo dx +3 уко ylo) | y (D)=1 3) xy+y=xy² -4) y" + wy = a Shwx y(0)=0 y(0)=1 W-constant ==