A certain spring-mass system satisfies the initial value
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A certain spring-mass system satisfies the initial value problem u?? + 0.25u? + u = kg(t), u(0) = 0, u? (0) = 0, where g
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A certain spring-mass system satisfies the initial value problem u?? + 0.25u? + u = kg(t), u(0) = 0, u? (0) = 0, where g(t) is equal to 1 over the time interval 3 2 < t < 5 2 , is zero elsewhere, and k > 0 is a parameter. (a) Sketch the graph of g(t). (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 > ?.
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