Consider the initial value problem y_ = t2 + ey , y(0) = 0. a. Let y = (t) be the

Chapter 8, Problem 2

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Consider the initial value problem y_ = t2 + ey , y(0) = 0. a. Let y = (t) be the solution of initial value problem (27). Further, let y = 1(t) be the solution of y_ = 1 + ey , y(0) = 0, (27) and let y = 2(t) be the solution of y_ = ey , y(0) = 0. (28) Show that 2(t) (t) 1(t) (29) on some interval, contained in 0 t 1, where all three solutions exist. b. Determine 1(t) and 2(t). Then show that (t) for some t between t = ln 2 = 0.69315 and t = 1. c. Solve the differential equations y_ = ey and y_ = 1 + ey , respectively, with the initial condition y(0.9) = 3.4298. Use the results to show that (t) when t = 0.932.