Assignment 9 for MATH 3270A Ordinary Differential Equations (November 13, 2014) 1. For the following ODEs, find the eigenvalues and eigenvectors, and classify the critical point (0, 0) as to type and determine whether it is stable, asymptotically stable, or unstable (a) (c) 3 −2 2 −2 ! 0 2 −5 0 2 ! 0 x = x = x, x, 1 −5 1 −3 ! 0 1 −2 1 −1 ! 0 (b) x = (d) x = 0 2. Transform the following problems into the form x = (a) 0 x = 3 −2 4 −2 ! x, 0 (b) x = 2 −2 3 −2 x, x. λ µ −µ λ ! x: ! x. 3. Determine the critical points for each of the following systems (a) (b) (c) (d) dx dt dx dt dx dt dx dt dy = y(1 − x − y) dt dy 1 1 3 = x − x2 − xy, = y − y 2 − xy dt 2 4 4 dy = y, = µ(1 − x2 )y − x, µ > 0 dt dy = y + x(1 − x2 − y 2 ), = −x + y(1 − x2 − y 2 ) dt = x(2x − 3y), 4. Show that the trajectories of the nonlinear undamped pendulum equation d2 θ g + sin θ = 0 dt2 l are given by g y2 (1 − cos x) + =c l 2 where x = θ, y = dθ . dt 1 5. Prove that if a trajectory starts at a noncritical point of the system dx = f (x) dt then it can not reach a critical point in a finite length of time. 6. Assuming that the trajectory corresponding to a solution x = φ(t), y = ψ(t), −∞ < t < +∞, of an autonomous system is closed, show that the solution must be periodic. 7. Write the following spring-mass system as a system of equations by introducing du x = u, u = dt d2 u du m 2 + c + ku = 0 dt dt Find out the critical point and analyze the stability of the critical point. 8. Show that the system is almost linear and (0, 0) is a stable critical pint of the system dx = −x − xy 2 , dt dy = −y − x2 y . dt 9. Show that the system is almost linear and (0, 0) is an asymptotically stable critical point of the system 1 dy dx = − x3 + 2xy 2 , = −y 3 dt 2 dt 10. Determine all real critical points of the following system of equations (a) (b) dx dy = x + y2, =x+y dt dt dx dy = 1 − y, = x2 − y 2 dt dt 11. Consider the following problem. Determine the eigenvalues and critical points, classify the type of critical point and determine whether it is stable, asymptotically stable, or unstable. ! 1 0 x x = −1 here is a real number. 2