Solved: In each of 16 and 17, solve the given system of equations by the method of of

In each of Problems 1 through 4: G a. Draw a direction field and sketch a few trajectories. b. Express the general solution of the given system of equations in terms of real-valued functions. c. Describe the behavior of the solutions as t .x_ = _1 4 1 1 _ x

In each of Problems 1 through 4: G a. Draw a direction field and sketch a few trajectories. b. Express the general solution of the given system of equations in terms of real-valued functions. c. Describe the behavior of the solutions as t .x_ = _2 5 1 2 _ x

In each of Problems 1 through 4: G a. Draw a direction field and sketch a few trajectories. b. Express the general solution of the given system of equations in terms of real-valued functions. c. Describe the behavior of the solutions as t .x_ = _1 1 5 3 _ x

In each of Problems 1 through 4: G a. Draw a direction field and sketch a few trajectories. b. Express the general solution of the given system of equations in terms of real-valued functions. c. Describe the behavior of the solutions as t .x_ = _ 1 2 5 1 _ X

In each of Problems 5 and 6, express the general solution of the given system of equations in terms of real-valued functions.x_ = 1 0 0 2 1 2 3 2 1 x

In each of Problems 5 and 6, express the general solution of the given system of equations in terms of real-valued functions.x_ = 3 0 2 1 1 0 2 1 0 x

In each of Problems 7 and 8, find the solution of the given initial-value problem. Describe the behavior of the solution as t .x_ = _1 5 1 3 _ x, x(0) = _11 _

In each of Problems 7 and 8, find the solution of the given initial-value problem. Describe the behavior of the solution as t .x_ = _3 2 1 1 _ x, x(0) = _ 1 2 _

In each of Problems 9 and 10: a. Find the eigenvalues of the given system. G b. Choose an initial point (other than the origin) and draw the corresponding trajectory in the x1x2-plane. G c. For your trajectory in part b, draw the graphs of x1 versus t and of x2 versus t. G d. For your trajectory in part b, draw the corresponding graph in three-dimensional tx1x2-space. Note that the projections of this plot onto each of the coordinate planes should produce the three plots produced in parts b and c.x_ = 34 2 1 5 4 x 1

In each of Problems 9 and 10: a. Find the eigenvalues of the given system. G b. Choose an initial point (other than the origin) and draw the corresponding trajectory in the x1x2-plane. G c. For your trajectory in part b, draw the graphs of x1 versus t and of x2 versus t. G d. For your trajectory in part b, draw the corresponding graph in three-dimensional tx1x2-space. Note that the projections of this plot onto each of the coordinate planes should produce the three plots produced in parts b and c.x_ = 4 5 2 1 6 5 x 1

In each of Problems 11 through 15, the coefficient matrix contains a parameter . In each of these problems: a. Determine the eigenvalues in terms of . b. Find the bifurcation value or values of where the qualitative nature of the phase portrait for the system changes. G c. Draw a phase portrait for a value of slightly below, and for another value slightly above, each bifurcation value.x_ = _ 1 1 _ x 1

In each of Problems 11 through 15, the coefficient matrix contains a parameter . In each of these problems: a. Determine the eigenvalues in terms of . b. Find the bifurcation value or values of where the qualitative nature of the phase portrait for the system changes. G c. Draw a phase portrait for a value of slightly below, and for another value slightly above, each bifurcation value.x_ = _0 5 1 _ x 1

In each of Problems 11 through 15, the coefficient matrix contains a parameter . In each of these problems: a. Determine the eigenvalues in terms of . b. Find the bifurcation value or values of where the qualitative nature of the phase portrait for the system changes. G c. Draw a phase portrait for a value of slightly below, and for another value slightly above, each bifurcation value.x_ = 54 3 4 54 x 1

In each of Problems 11 through 15, the coefficient matrix contains a parameter . In each of these problems: a. Determine the eigenvalues in terms of . b. Find the bifurcation value or values of where the qualitative nature of the phase portrait for the system changes. G c. Draw a phase portrait for a value of slightly below, and for another value slightly above, each bifurcation value.x_ = _1 1 1 _ x 1

In each of Problems 11 through 15, the coefficient matrix contains a parameter . In each of these problems: a. Determine the eigenvalues in terms of . b. Find the bifurcation value or values of where the qualitative nature of the phase portrait for the system changes. G c. Draw a phase portrait for a value of slightly below, and for another value slightly above, each bifurcation value.x_ = _4 8 6 _ x 1

In each of Problems 16 and 17, solve the given system of equations by the method of Problem 13 of Section 7.5. Assume that t > 0.tx_ = _1 1 2 1 _ x 1

In each of Problems 16 and 17, solve the given system of equations by the method of Problem 13 of Section 7.5. Assume that t > 0.tx_ = _2 5 1 2 _ x 1

In each of Problems 18 and 19: a. Find the eigenvalues of the given system. G b. Choose an initial point (other than the origin) and draw the corresponding trajectory in the x1x2-plane. Also draw the trajectories in the x1x3- and x2x3-planes. G c. For the initial point in part b, draw the corresponding trajectory in x1x2x3-space.x_ = 1 4 1 0 1 1 4 0 0 0 1 4 x 1

In each of Problems 18 and 19: a. Find the eigenvalues of the given system. G b. Choose an initial point (other than the origin) and draw the corresponding trajectory in the x1x2-plane. Also draw the trajectories in the x1x3- and x2x3-planes. G c. For the initial point in part b, draw the corresponding trajectory in x1x2x3-space.x_ = 1 4 1 0 1 1 4 0 0 0 1 10 x 2

Consider the electric circuit shown in Figure 7.6.6. Suppose that R1 = R2 = 4 , C = 1 2 F, and L = 8 H. a. Show that this circuit is described by the system of differential equations d dt _I V _ = 1 2 1 8 2 12 _I V _, (32) where I is the current through the inductor and V is the voltage drop across the capacitor. Hint: See Problem 18 of Section 7.1. b. Find the general solution of equations (32) in terms of realvalued functions. c. Find I (t) and V(t) if I (0) = 2 A and V(0) = 3 V. d. Determine the limiting values of I (t) and V(t) as t . Do these limiting values depend on the initial conditions? R1 R2 L C FIGURE 7.6.6 The circuit in Problem 20. 2

The electric circuit shown in Figure 7.6.7 is described by the system of differential equations d dt _I V _ = 0 1L 1C 1 RC _I V _, (33) where I is the current through the inductor and V is the voltage drop across the capacitor. These differential equations were derived in Problem 16 of Section 7.1. a. Show that the eigenvalues of the coefficient matrix are real and different if L > 4R2C; show that they are complex conjugates if L < 4R2C. b. Suppose that R = 1 , C = 12 F, and L = 1 H. Find the general solution of the system (33) in this case. c. Find I (t) and V(t) if I (0) = 2 A and V(0) = 1 V. d. For the circuit of part b, determine the limiting values of I (t) and V(t) as t . Do these limiting values depend on the initial conditions? C L R FIGURE 7.6.7 The circuit in Problem 21. 2

In this problem we indicate how to show that u(t) and v(t), as given by equations (17), are linearly independent. Let r1 = + i and r1 = i be a pair of conjugate eigenvalues of the coefficient matrix A of equation (1); let (1) = a + ib and (1) = a ib be the corresponding eigenvectors. Recall that it was stated in Section 7.3 that two different eigenvalues have linearly independent eigenvectors, so if r1 _= r1, then (1) and (1) are linearly independent. a. First we show that a and b are linearly independent. Consider the equation c1a+c2b = 0. Express a and b in terms of (1) and (1) , and then show that (c1 ic2) (1) + (c1 + ic2) (1) = 0. b. Show that c1ic2 = 0 and c1+ic2 = 0 and then that c1 = 0 and c2 = 0. Consequently, a and b are linearly independent. c. To show that u(t) and v(t) are linearly independent, consider the equation c1u(t0)+c2v(t0) = 0, where t0 is an arbitrary point. Rewrite this equation in terms of a and b, and then proceed as in part b to show that c1 = 0 and c2 = 0. Hence u(t) and v(t) are linearly independent at the arbitrary point t0. Therefore, they are linearly independent at every point and on every interval. 2

A mass m on a spring with constant k satisfies the differential equation (see Section 3.7) mu__ + ku = 0, where u(t) is the displacement at time t of the mass from its equilibrium position. a. Let x1 = u, x2 = u_, and show that the resulting system is x _ = 0 1 k m 0 _ x. b. Find the eigenvalues of the matrix for the system in part a. c. Sketch several trajectories of the system. Choose one of your trajectories, and sketch the corresponding graphs of x1 versus t and x2 versus t. Sketch both graphs on one set of axes. d. What is the relation between the eigenvalues of the coefficient matrix and the natural frequency of the spring-mass system? 2

Consider the two-mass, three-spring system of Example 3 in the text. Instead of converting the problem into a system of four first-order equations, we indicate here how to proceed directly from equations (22). a. Show that equations (22) can be written in the form x __ = 2 3 2 4 3 3 x = Ax. (34) b. Assume that x = ert and show that (A r2I) = 0. Note that r2 (rather than r) is an eigenvalue of A corresponding to an eigenvector . c. Find the eigenvalues and eigenvectors of A. d. Write down expressions for x1 and x2. There should be four arbitrary constants in these expressions. e. By differentiating the results from part d, write down expressions for x_ 1 and x_ 2. Your results from parts d and e should agree with equation (31) in the text. 2

Consider the two-mass, three-spring system whose equations of motion are equations (22). Let m1 = 1, m2 = 4/3, k1 = 1, k2 = 3, and k3 = 4/3. a. As in Example 3, convert the system to four first-order equations of the form y_ = Ay. Determine the coefficient matrix A. b. Find the eigenvalues and eigenvectors of A. c. Write down the general solution of the system. G d. Describe the fundamental modes of vibration. For each fundamental mode, draw graphs of y1 versus t and y2 versus t. Also draw the corresponding trajectories in the y1 y3- and y2 y4- planes. G e. Consider the initial conditions y(0) = (2, 1, 0, 0)T . Evaluate the arbitrary constants in the general solution in part c. What is the period of the motion in this case? Plot graphs of y1 versus t and y2 versus t. Also plot the corresponding trajectories in the y1 y3- and y2 y4-planes. Be sure you understand how the trajectories are traversed for a full period. G f. Consider other initial conditions of your own choice, and plot graphs similar to those requested in part e.