Now solved: For each of the systems in 1 through 12:(a) Find the eigenvalues and

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  3 2 2 2  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  5 1 3 1 x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  2 1 3 2  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  1 4 4 7  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  1 5 1 3  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  2 5 1 2  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  3 2 4 1  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  1 1 0 0.25 x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  3 4 1 1  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  1 2 5 1  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  1 0 0 1  x

For each of the systems in Problems 1 through 12: (a) Find the eigenvalues and eigenvectors. (b) Classify the critical point(0, 0) as to type, and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane, and also sketch some typical graphs of x1 versus t. (d) Use a computer to plot accurately the curves requested in part (c). dx dt =  2 5 2 9 5 1  x

In each of Problems 13 through 16, determine the critical point x = x0, and then classify its type and examine its stability by making the transformation x = x0 + u. dx dt =  1 1 1 1  x  2 0 

In each of Problems 13 through 16, determine the critical point x = x0, and then classify its type and examine its stability by making the transformation x = x0 + u. dx dt =  2 1 1 2  x +  2 1 

In each of Problems 13 through 16, determine the critical point x = x0, and then classify its type and examine its stability by making the transformation x = x0 + u. dx dt =  1 1 2 1  x +  1 5 

In each of Problems 13 through 16, determine the critical point x = x0, and then classify its type and examine its stability by making the transformation x = x0 + u. dx dt =  0 0  x +   ; , , , > 0

The equation of motion of a springmass system with damping (see Section 3.7) is md2u dt2 + c du dt + ku = 0, where m, c, and k are positive. Write this second order equation as a system of two first order equations for x = u, y = du/dt. Show that x = 0, y = 0 is a critical point,and analyze the nature and stability of the critical point as a function of the parameters m, c, and k. A similar analysis can be applied to the electric circuit equation (see Section 3.7) L d2I dt2 + R dI dt + 1 C I = 0.

Consider the system x = Ax, and suppose that A has one zero eigenvalue. (a) Show that det(A) = 0. (b) Show that x = 0 is a critical point and that,in addition, every point on a certain straight line through the origin is also a critical point. (c) Let r1 = 0 and r2 = 0, and let (1) and (2) be corresponding eigenvectors. Show that the trajectories are as indicated in Figure 9.1.8. What is the direction of motion on the trajectories?

In this problem we indicate how to show that the trajectories are ellipses when the eigenvalues are pure imaginary. Consider the system  x y  =  a11 a12 a21 a22 x y  . (i) (a) Show that the eigenvalues of the coefficient matrix are pure imaginary if and only if a11 + a22 = 0, a11a22 a12a21 > 0. (ii) (b) The trajectories of the system (i) can be found by converting Eqs. (i) into the single equation dy dx = dy/dt dx/dt = a21x + a22y a11x + a12y . (iii) Use the first of Eqs. (ii) to show that Eq. (iii) is exact. (c) By integrating Eq. (iii), show that a21x2 + 2a22xy a12y2 = k, (iv) where k is a constant. Use Eqs. (ii) to conclude that the graph of Eq. (iv) is always an ellipse. Hint: What is the discriminant of the quadratic form in Eq. (iv)?

Consider the linear system dx/dt = a11x + a12y, dy/dt = a21x + a22y, where a11, a12, a21, and a22 are real constants. Let p = a11 + a22, q = a11a22 a12a21, and  = p2 4q. Observe that p and q are the trace and determinant, respectively, of the coefficient matrix of the given system. Show that the critical point (0, 0) is a (a) Node if q > 0 and  0; (b) Saddle point if q < 0; (c) Spiral point if p = 0 and  < 0; (d) Center if p = 0 and q > 0. Hint: These conclusions can be reached by studying the eigenvalues r1 and r2. It may also be helpful to establish, and then to use, the relations r1r2 = q and r1 + r2 = p.

Continuing Problem 20, show that the critical point (0, 0) is (a) Asymptotically stable if q > 0 and p < 0; (b) Stable if q > 0 and p = 0; (c) Unstable if q < 0 or p > 0. The results of Problems 20 and 21 are summarized visually in Figure 9.1.9.

In this problem we illustrate how a 2 2 system with eigenvalues i can be transformed into the system (11). Consider the system in Problem 12: x =  2 2.5 1.8 1  x = Ax. (i) (a) Show that the eigenvalues of this system are r1 = 0.5 + 1.5i and r2 = 0.5 1.5i. (b) Show that the eigenvector corresponding to r1 can be chosen as (1) =  5 3 3i  =  5 3  + i  0 3  . (ii) (c) Let P be the matrix whose columns are the real and imaginary parts of (1) . Thus P =  5 0 3 3  . (iii) Let x = Py and substitute for x in Eq. (i). Show that y = (P1 AP)y. (iv) (d) Find P1 and show that P1 AP =  0.5 1.5 1.5 0.5  . (v) Thus Eq. (v) has the form of Eq. (11).