Differential Equations

y' (t) = -(1 + sint)y(t) y(0) = 1 (1) (a) (4 pts) Find the solution to the initial value problem Eq.1 (b) (2 pts) Let o(t) be the solution you found. Show that v(t) = o(t + 27) satisfies the differential equation but not the initial condition. (c) (2 pts) Find the limit of $(2mm) as m -+ co. (d) (2 pts) Noting that in this example o(t) # o(t+27), generalize your result in (b) and show that if u(t) is a solution to u' + f(t)u = 0 where f(t) = f(t + 27), then so too is v(t) = u(t + 27). Hint: do not try to solve for u(t). Part B: (16 pts) Let dry at 2 = f(t)y (2) where f(t + 27) = f(t). Let yi(t) and y2(t) be linearly independent functions that satisfy Eq. 2. Take it as a given that yi(t + 27) and y2(t + 27) are also solutions. (It follows from the same argument as in Part A(d)). Any solution y(t) = a yi(t) + by2(t) and in particular y1 (t + 27) = a y1(t) + by2(t) y2 ( t + 27) = cy1(t) + d yz(t) (3) (a) (2 pts) Let (t) = (y1(t) yz(t))T. Rewrite Eqs. 3 as a matrix equa- tion. (b) (14 pts) Find $(2mm) in the limit m - co for case where a = 4, b = -5, c = 2, d = -3 and y1 (0) = 1, y2(0) = -1