Consider the initial value problem y + 1 3 y + 4y = fk(t), y(0) = 0, y (0) = 0, where
Chapter 6, Problem 18(choose chapter or problem)
Consider the initial value problem y + 1 3 y + 4y = fk(t), y(0) = 0, y (0) = 0, where fk(t) = 1/2k, 4 k t < 4 + k 0, 0 t < 4 k and t 4 + k and 0 < k < 4. (a) Sketch the graph of fk(t). Observe that the area under the graph is independent of k. If fk(t) represents a force, this means that the product of the magnitude of the force and the time interval during which it acts does not depend on k. (b) Write fk(t) in terms of the unit step function and then solve the given initial value problem. (c) Plot the solution for k = 2, k = 1, and k = 1 2 . Describe how the solution depends on k.
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