Modify the problem in Example 1 of this section by repla

Modify the problem in Example 1 of this section by replacing the given forcing function g(t) by f(t) = u5(t)(t ? 5) ? u5+k(t)(t ? 5 ? k) k . (a) Sketch the graph of f(t) and describe how it depends on k. For what value of k is f(t) identical to g(t) in the example? (b) Solve the initial value problem y?? + 4y = f(t), y(0) = 0, y? (0) = 0. (c) The solution in part (b) depends on k, but for sufficiently large t the solution is always a simple harmonic oscillation about y = 1 4 . Try to decide how the amplitude of this eventual oscillation depends on k. Then confirm your conclusion by plotting the solution for a few different values of k.