Given that x = (t), y = (t) is a solution of the autonomous system dx/dt = F(x, y)

In each of Problems 1 through 4 sketch the trajectory corresponding to the solution satisfying the specified initial conditions, and indicate the direction of motion for increasing t.dx/dt = x, dy/dt = 2y; x(0) = 4, y(0) = 2

In each of Problems 1 through 4 sketch the trajectory corresponding to the solution satisfying the specified initial conditions, and indicate the direction of motion for increasing t.. dx/dt = x, dy/dt = 2y; x(0) = 4, y(0) = 2 and x(0) = 4, y(0) = 0

In each of Problems 1 through 4 sketch the trajectory corresponding to the solution satisfying the specified initial conditions, and indicate the direction of motion for increasing t.dx/dt = y, dy/dt = x; x(0) = 4, y(0) = 0 and x(0) = 0, y(0) = 4

In each of Problems 1 through 4 sketch the trajectory corresponding to the solution satisfying the specified initial conditions, and indicate the direction of motion for increasing t.dx/dt = ay, dy/dt = bx, a > 0, b > 0; x(0) = a, y(0) = 0

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.dx/dt = x xy, dy/dt = y + 2xy

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.dx/dt = 1 + 2y, dy/dt = 1 3x2

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.dx/dt = (2 + y)(x + y), dy/dt = y(1 x)

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.. dx/dt = (2 + x)(y x), dy/dt = y(2 + x x2)

dx/dt = x + 2xy, dy/dt = y x2 y2

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.dx/dt = (2 + x)(y x), dy/dt = (4 x)(y + x)

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.dx/dt = x(2 x y), dy/dt = x + 3y 2xy

For each of the systems in Problems 5 through 16: (a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and phase portrait for the system. (c) From the plot(s) in part (b), determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. (d) Describe the basin of attraction for each asymptotically stable critical point.dx/dt = x(2 x y), dy/dt = (1 y)(2 + x)

In each of Problems 17 through 24: (a) Find an equation of the form H(x, y) = c satisfied by the trajectories. (b) Plot several level curves of the function H. These are trajectories of the given system. Indicate the direction of motion on each trajectory.dx/dt = 2y, dy/dt = 8x

In each of Problems 17 through 24: (a) Find an equation of the form H(x, y) = c satisfied by the trajectories. (b) Plot several level curves of the function H. These are trajectories of the given system. Indicate the direction of motion on each trajectory.dx/dt = 2y, dy/dt = 8x

In each of Problems 17 through 24: (a) Find an equation of the form H(x, y) = c satisfied by the trajectories. (b) Plot several level curves of the function H. These are trajectories of the given system. Indicate the direction of motion on each trajectory.

In each of Problems 17 through 24: (a) Find an equation of the form H(x, y) = c satisfied by the trajectories. (b) Plot several level curves of the function H. These are trajectories of the given system. Indicate the direction of motion on each trajectory.

In each of Problems 17 through 24: (a) Find an equation of the form H(x, y) = c satisfied by the trajectories. (b) Plot several level curves of the function H. These are trajectories of the given system. Indicate the direction of motion on each trajectory.dx/dt = x + y + x2, dy/dt = y 2xy

In each of Problems 17 through 24: (a) Find an equation of the form H(x, y) = c satisfied by the trajectories. (b) Plot several level curves of the function H. These are trajectories of the given system. Indicate the direction of motion on each trajectory.dx/dt = 2x2y 3x2 4y, dy/dt = 2xy2 + 6xy

In each of Problems 17 through 24: (a) Find an equation of the form H(x, y) = c satisfied by the trajectories. (b) Plot several level curves of the function H. These are trajectories of the given system. Indicate the direction of motion on each trajectory.Undamped pendulum: dx/dt = y, dy/dt = sin x

In each of Problems 17 through 24: (a) Find an equation of the form H(x, y) = c satisfied by the trajectories. (b) Plot several level curves of the function H. These are trajectories of the given system. Indicate the direction of motion on each trajectory.Duffings2 equations: dx/dt = y, dy/dt = x + (x3/6)

Given that x = (t), y = (t) is a solution of the autonomous system dx/dt = F(x, y), dy/dt = G(x, y) for < t < , show that x = (t) = (t s), y = (t) = (t s) is a solution for + s < t < + s for any real number s.

Prove that for the system dx/dt = F(x, y), dy/dt = G(x, y) there is at most one trajectory passing through a given point (x0, y0). Hint: Let C0 be the trajectory generated by the solution x = 0(t), y = 0(t), with 0(t0) = x0, 0(t0) = y0, and let C1 be the trajectory generated by the solution x = 1(t), y = 1(t), with 1(t1) = x0, 1(t1) = y0. Use the fact that the system is autonomous, and also the existence and uniqueness theorem, to show that C0 and C1 are the same.

Prove that if a trajectory starts at a noncritical point of the system dx/dt = F(x, y), dy/dt = G(x, y), then it cannot reach a critical point (x0, y0) in a finite length of time. Hint: Assume the contrary; that is, assume that the solution x = (t), y = (t) satisfies (a) = x0, (a) = y0. Then use the fact that x = x0, y = y0 is a solution of the given system satisfying the initial condition x = x0, y = y0 at t = a.

Assuming that the trajectory corresponding to a solution x = (t), y = (t), < t < , of an autonomous system is closed, show that the solution is periodic. Hint: Since the trajectory is closed, there exists at least one point (x0, y0) such that (t0) = x0,(t0) = y0 and a number T > 0 such that (t0 + T) = x0,(t0 + T) = y0. Show that x = (t) = (t + T) and y = (t) = (t + T)is a solution, and then use the existence and uniqueness theorem to show that (t) = (t) and (t) = (t) for all t.