A certain springmass system satisfies the initial value problem u + 1 4u + u = kg(t)

Chapter 6, Problem 16

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A certain springmass system satisfies the initial value problem u + 1 4u + u = kg(t), u(0) = 0, u (0) = 0, where g(t) = u3/2(t) u5/2(t) and k > 0 is a parameter. (a) Sketch the graph of g(t). Observe that it is a pulse of unit magnitude extending over one time unit. (b) Solve the initial value problem. (c) Plot the solution for k = 1/2, k = 1, and k = 2. Describe the principal features of the solution and how they depend on k. (d) Find, to two decimal places, the smallest value of k for which the solution u(t) reaches the value 2. (e) Suppose k = 2. Find the time after which |u(t)| < 0.1 for all t >