Answer Problems 1–10 without referring back to the text. Fill in the blank or answer true/false.

1. The second-order differential equation x″ + f(x′) + g(x) = 0 can be written as a plane autonomous system.
2. If X = X(t) is a solution of a plane autonomous system and X(t₁) = X(t₂) for t₁ ≠ t₂, then X(t) is a periodic solution.
3. If the trace of the matrix A is 0 and det A ≠ 0, then the critical point (0, 0) of the linear system X′ = AX may be classified as .
4. If the critical point (0, 0) of the linear system X′ = AX is a stable spiral point, then the eigenvalues of A are .
5. If the critical point (0, 0) of the linear system X′ = AX is a saddle point and X = X(t) is a solution, then lim_t→∞ X(t) does not exist.
6. If the Jacobian matrix A = g′(X₁) at a critical point of a plane autonomous system has positive trace and determinant, then the critical point X₁ is unstable.
7. It is possible to show that a nonlinear plane autonomous system has periodic solutions using linearization.
8. All solutions to the pendulum equation sin θ = 0 are periodic.
9. If a simply connected region R contains no critical points of a plane autonomous system, then there are no periodic solutions in R.
10. If a plane autonomous system has no critical points in an annular invariant region R, then there is at least one periodic solution in R.
11. Solve the following nonlinear plane autonomous system by switching to polar coordinates, and describe the geometric behavior of the solution that satisfies the given initial condition.

    

12. Discuss the geometric nature of the solutions to the linear system X′ = AX given the general solution.
    1. X(t) = c₁e^–t + c₂e^–2t
    2. X(t) = c₁e^–t + c₂e^2t
13. Classify the critical point (0, 0) of the given linear system by computing the trace τ and determinant Δ.
    1. x′ = –3x + 4y
       y′ = –5x + 3y
    2. x′ = –3x + 2y
       y′ = –2x + y
14. Find and classify (if possible) the critical points of the plane autonomous system

    x′ = x + xy – 3x²

       y′ = 4y – 2xy – y².

    Does this system have any periodic solutions in the first quadrant?

15. Classify the critical point (0, 0) of the plane autonomous system corresponding to the nonlinear second-order differential equation

    x″ + µ(x² – 1)x′ + x = 0

    where µ is a real constant.

16. Without solving explicitly, classify (if possible) the critical points of the autonomous first-order differential equation x′ = (x² – 1)e^–x/2 as asymptotically stable or unstable.
17. Use the phase-plane method to show that the solutions of the nonlinear second-order differential equation

    

    that satisfy x(0) = x₀ and x′(0) = 0 are periodic.

18. In Section 3.8 we assumed that the restoring force F of the spring satisfied Hooke’s law F = ks, where s is the elongation of the spring and k is a positive constant of proportionality. If we replace this assumption with the nonlinear law F = ks³, then the new differential equation for damped motion becomes mx″ = –βx′ – k(s + x)³ + mg, where ks³ = mg. The system is called overdamped when (0, 0) is a stable node and is called underdamped when (0, 0) is a stable spiral point. Find new conditions on m, k, and β that will lead to overdamping and underdamping.
19. Show that the plane autonomous system

    x′ = 4x + 2y – 2x²

    y′ = 4x – 3y + 4xy

    has no periodic solutions.

20. Use the Poincaré–Bendixson theorem to show that the plane autonomous system

    x′ = ϵx + y – x(x² + y²)

      y′ = –x + ϵy – y(x² + y²)

    has at least one periodic solution when ϵ > 0. What occurs when ϵ < 0?

21. The rod of a pendulum is attached to a movable joint at point P and rotates at an angular speed of ω (radians/s) in the plane perpendicular to the rod. See FIGURE 11.R.1. As a result, the bob of the pendulum experiences an additional centripetal force and the new differential equation for θ becomes

    ml = ω²ml sin θ cos θ – mg sin θ – β .

    1. Establish that there are no periodic solutions.
    2. If ω² < g/l, show that (0, 0) is a stable critical point and is the only critical point in the domain –π < θ < π. Describe what occurs physically when θ(0) = θ₀, θ′(0) = 0, and θ₀ is small.
    3. If ω² > g/l, show that (0, 0) is unstable and there are two additional stable critical points (±, 0) in the domain –π < θ < π. Describe what occurs physically when θ(0) = θ₀, θ′(0) = 0, and θ₀ is small.
    4. Determine under what conditions the critical points in parts (a) and (b) are stable spiral points.

       

22. The nonlinear second-order differential equation

    x″ – 2kx′ + c(x′)³ + ω²x = 0

    arises in modeling the motion of an electrically driven tuning fork. See FIGURE 11.R.2, where k = c = 0.1 and ω = 1. Assume that this differential equation possesses a Type I invariant region that contains (0, 0). Show that there is at least one periodic solution.