Consider the initial value problem y?? + y = fk(t), y(0)
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Consider the initial value problem y?? + y = fk(t), y(0) = 0, y? (0) = 0, where fk(t) = [u3?k(t) ? u3+k(t)]?2k with 0 <
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Consider the initial value problem y?? + y = fk(t), y(0) = 0, y? (0) = 0, where fk(t) = [u3?k(t) ? u3+k(t)]?2k with 0 < k ? 1. (a) Find the solution y = ?(t, k) of the initial value problem. (b) Calculate limk?0 ?(t, k) from the solution found in part (a). (c) Observe that limk?0 fk(t) = ?(t ? 3). Find the solution ?0(t) of the given initial value problem, with f k(t) replaced by ?(t ? 3). Is it true that ?0(t) = limk?0 ?(t, k)? (d) Plot ?(t, 1 2 ), ?(t, 1 4 ), and ?0(t) on the same axes. Describe the relation between ?(t, k) and ?0(t). Problems 17 through 22 deal with the effect of a sequence of impulses on an undamped oscillator. Suppose that y?? + y = f(t), y(0) = 0, y? (0) = 0. For each of the following choices for f(t): (a) Try to predict the nature of the solution without solving the problem. (b) Test your prediction by finding the solution and drawing its graph. (c) Determine what happens after the sequence of impulses ends.
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