# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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PROBLEMS In each of Problems 1 through 4, verify that (0, 0) is a critical point, show that the system is

I almost linear, and discuss the type and stability of the critical point (0, 0) by examining the corresponding linear system.

1. dx/dt = x - y, dy/dt = x - 2y + x2

2. dx/dt = —x + y + 2xy, dy/dt =-4x - y + x2 - y2

3. dx/dt = (1 + x) sin y, dy/dt = 1 - x - cos y

4. dx/dt = x + y2, dy/dt = x + y

In each of Problems 5 through 16:

(a) Determine all critical points of the given system of equations.

(b) Find the corresponding linear system near each critical point.

(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

(d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.

? 5. dx/dt = (2 + x)(y - x), dy/dt = (4 - x)(y + x)

? 6. dx/dt = x - x2 - xy, dy/dt = 3y - xy - 2y2

? 7. dx/dt = 1 - y, dy/dt = x2 - y2

? 8. dx/dt = x - x2 - xy, dy/dt = 2y - 1 y2 - 3xy

? 9. dx/dt = — (x - y)(1 - x - y), dy/dt = x(2 + y)

? 10. dx/dt = x + x2 + y2, dy/dt = y - xy

? 11. dx/dt = 2x + y + xy3, dy/dt = x - 2y - xy

488

Chapter 9. Nonlinear Differential Equations and Stability

? 12. dx/dt = (1 + x) siny, dy/dt = 1 - x - cosy

? 13. dx/dt = x - y2, dy/dt = y - x2 ? 14. dx/dt = 1 - xy, dy/dt = x - y3

? 15. dx/dt = -2x - y - x(x2 + y2), dy/dt = x - y + y(x2 + y2)

? 16. dx/dt = y + x(1 - x2 - y2), dy/dt =-x + y(1 - x2 - y2)

17. Consider the autonomous system

(a) Show that the critical point (0, 0) is a saddle point.

(b) Sketch the trajectories for the corresponding linear system by integrating the equation for dy/dx. Show from the parametric form of the solution that the only trajectory on which x ^ 0, y ^ 0as t ^to is y = —x.

(c) Determine the trajectories for the nonlinear system by integrating the equation for dy/dx. Sketch the trajectories for the nonlinear system that correspond to y = — x and y = x for the linear system.

18. Consider the autonomous system

(a) Show that the critical point (0, 0) is a saddle point.

(b) Sketch the trajectories for the corresponding linear system and show that the trajectory

for which x ^ 0, y ^ 0 as t ^ to is given by x = 0.

(c) Determine the trajectories for the nonlinear system for x = 0 by integrating the equa-

tion for dy/dx. Show that the trajectory corresponding to x = 0 for the linear system is unaltered, but that the one corresponding to y = 0 is y = x3/5. Sketch several of the trajectories for the nonlinear system.

? 19. The equation of motion of an undamped pendulum is d29/dt2 + a>2 sin 9 = 0, where a>2 = g/ L .Let x = 9, y = d9/dt to obtain the system of equations

(a) Show that the critical points are (±nn, 0), n = 0, 1, 2,..., and that the system is almost linear in the neighborhood of each critical point.

(b) Show that the critical point (0, 0) is a (stable) center of the corresponding linear system. Using Theorem 9.3.2 what can be said about the nonlinear system? The situation is similar at the critical points (±2nn, 0), n = 1, 2, 3,.... What is the physical interpretation of these critical points?

(c) Show that the critical point (n, 0) is an (unstable) saddle point of the corresponding linear system. What conclusion can you draw about the nonlinear system? The situation is similar at the critical points [±(2n — 1)n, 0], n = 1, 2, 3,.... What is the physical interpretation of these critical points?

(d) Choose a value for rn2 and plot a few trajectories of the nonlinear system in the neighborhood of the origin. Can you now draw any further conclusion about the nature of the critical point at (0, 0) for the nonlinear system?

(e) Using the value of a>2 from part (d) draw a phase portrait for the pendulum. Compare your plot with Figure 9.3.5 for the damped pendulum.

20. (a) By solving the equation for dy/dx, show that the equation of the trajectories of the undamped pendulum of Problem 19 can be written as

dx/dt = y, dy/dt = x + 2x3.

dx/dt = x, dy/dt = —2 y + x3.

dx/dt = y, dy/dt = —a>2 sin x.

2 y2 + «2(1 — cos x) = c,

(i)

where c is a constant of integration.

(b) Multiply Eq. (i) by mL2. Then express the result in terms of 9 to obtain

(ii)

9.3 Almost Linear Systems

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(c) Show that the first term in Eq. (ii) is the kinetic energy of the pendulum and that the second term is the potential energy due to gravity. Thus the total energy E of the pendulum is constant along any trajectory; its value is determined by the initial conditions.

? 21. The motion of a certain undamped pendulum is described by the equations

dx/dt — y, dy/dt — —4 sinx.

If the pendulum is set in motion with an angular displacement A and no initial velocity, then the initial conditions are x(0) — A, y(0) — 0.

(a) Let A — 0.25 and plot x versus t. From the graph estimate the amplitude R and period T of the resulting motion of the pendulum.

(b) Repeat part (a) for A — 0.5, 1.0, 1.5, and 2.0.

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