Get answer: 17 through 22 deal with the effect of a sequence of impulses on an

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + 2y + 2y = (t ); y(0) = 1, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + 4y = (t ) (t 2); y(0) = 0, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + 3y + 2y = (t 5) + u10(t); y(0) = 0, y (0) = 1/2

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y y = 20(t 3); y(0) = 1, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + 2y + 3y = sin t + (t 3); y(0) = 0, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + 4y = (t 4); y(0) = 1/2, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + y = (t 2) cost; y(0) = 0, y (0) = 1

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + 4y = 2(t /4); y(0) = 0, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + y = u/2(t) + 3(t 3/2) u2(t); y(0) = 0, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution 2y + y + 4y = (t /6)sin t; y(0) = 0, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y + 2y + 2y = cost + (t /2); y(0) = 0, y (0) = 0

In each of Problems 1 through 12: (a) Find the solution of the given initial value problem. (b) Draw a graph of the solution y(4) y = (t 1); y(0) = 0, y (0) = 0, y(0) = 0, y(0) = 0

3. Consider again the system in Example 1 of this section, in which an oscillation is excited by a unit impulse at t = 5. Suppose that it is desired to bring the system to rest again after exactly one cyclethat is, when the response first returns to equilibrium moving in the positive direction. (a) Determine the impulse k(t t0) that should be applied to the system in order to accomplish this objective. Note that k is the magnitude of the impulse and t0 is the time of its application. (b) Solve the resulting initial value problem, and plot its solution to confirm that it behaves in the specified manner.

Consider the initial value problem y + y + y = (t 1), y(0) = 0, y (0) = 0, where is the damping coefficient (or resistance). (a) Let = 1 2 . Find the solution of the initial value problem and plot its graph. (b) Find the time t1 at which the solution attains its maximum value. Also find the maximum value y1 of the solution. (c) Let = 1 4 and repeat parts (a) and (b). (d) Determine how t1 and y1 vary as decreases. What are the values of t1 and y1 when = 0?

Consider the initial value problem y + y + y = k(t 1), y(0) = 0, y (0) = 0, where k is the magnitude of an impulse at t = 1, and is the damping coefficient (or resistance). (a) Let = 1 2 . Find the value of k for which the response has a peak value of 2; call this value k1. (b) Repeat part (a) for = 1 4 . (c) Determine how k1 varies as decreases. What is the value of k1 when = 0?

Consider the initial value problem y + y = fk(t), y(0) = 0, y (0) = 0, where fk(t) = [u4k(t) u4+k(t)]/2k with 0 < k 1. (a) Find the solution y = (t, k) of the initial value problem. (b) Calculate limk0+ (t, k) from the solution found in part (a). (c) Observe that limk0+ fk(t) = (t 4). Find the solution 0(t) of the given initial value problem with fk(t) replaced by (t 4). Is it true that 0(t) = lim k0+ (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. . f(t) = 20 k=1 (t k)

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. 18. f(t) = 20 k=1 (1)k+1(t k)

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. f(t) = 20 k=1 (t k/2)

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. f(t) = 20 k=1 (1)k+1(t k/2)

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. f(t) = 15 k=1 [t (2k 1)]

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. . f(t) = 40 k=1 (1)k+1(t 11k/4)

The position of a certain lightly damped oscillator satisfies the initial value problem y + 0.1y + y = 20 k=1 (1) k+1 (t k), y(0) = 0, y (0) = 0. Observe that, except for the damping term, this problem is the same as Problem 18. (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.

Proceed as in Problem 23 for the oscillator satisfying y + 0.1y + y = 15 k=1 [t (2k 1)], y(0) = 0, y (0) = 0. Observe that, except for the damping term, this problem is the same as Problem 21.

(a) By the method of variation of parameters, show that the solution of the initial value problem y + 2y + 2y = f(t); y(0) = 0, y (0) = 0 is y = t 0 e(t)f()sin(t ) d. (b) Show that if f(t) = (t ), then the solution of part (a) reduces to y = u(t)e(t) sin(t ). (c) Use a Laplace transform to solve the given initial value problem with f(t) = (t ), and confirm that the solution agrees with the result of part (b).