Questions 8, 10 and 14. Please! :D For the nonlinear system dx/dt = a- 4- xy and

Question

Questions 8, 10 and 14. Please! :D

For the nonlinear system dx/dt = a- 4- xy and dy/dt = y^2 - 2y, what is the linearized system for the equilibrium point at the origin? For the nonlinear system in Exercise 1, determine the type (sink, source, ...) of the equilibrium point at the origin. For the nonlinear system dx/dt = x^2 + sin 3x and dy/dt = 2y - sin x y, what is the linearized system for the equilibrium point at the origin? For the nonlinear system in Exercise 3. determine the type (sink, source, ...) of the equilibrium point at the origin. Sketch the nuIIclines for the system dx/dt = x - y and dy/dt = x^2 + y^2 - 2. Is the system dx/dt = 2xy + y^2 and dy/dt = x^2 + y^2 a Hamiltonian system? Is the system in Exercise 6 a gradient system? Describe three different possible long-term behaviors for a solution Y(f) of a two-dimensional system dY/dt = F(Y). Suppose that a two-dimensional system dY/dt = F(Y) has exactly one equilibrium point and that point is a source. Describe two different possible long-term behaviors for a solution with an initial condition that is near the equilibrium point. Suppose that the origin is an equilibrium point of a two-dimensional system and that zero is not an eigenvalue for the linearized system at the origin. Suppose also that dy/dt > 0 on the positive x-axis and dx/dt > 0 on the positive y-axis. What type of equilibrium point can the origin be? True-false: For Exercises 11-14. determine if the statement is true or false. If it is true, explain why. If it is false, provide a counterexample or an explanation. The equilibrium points of a first-order system occur at the intersection of the x- and y-nullclines. The x- and y-nullclines of a system are never identical. If (x_0, y_0) is an equilibrium point of the system dx/dt = f(x, y) and dy/dt = g(x, y), then the eigenvalues of the linearized system at (x_0, y_0) are the partial derivatives partial differential f/partial differential x and partial differential g/partial differential y evaluated at (x_0, y_0). For a system of the form dx/dt =/(x) and dy/dt = g(x, y) with f'(x) > 0 for all x, every equilibrium point is a source.

Explanation / Answer

8. dY/dt = F(Y)

i) F(Y) may be a constant k. This will give Y = kt + c. Then, Y will be directly proportional to t. Graph of Y and t will be a straight line.

ii) F(Y) may be linear function of Y, say aY + b. This will give aY + b = e^(at) + c. Then, rraph of Y and t will be exponential.

iii) F(Y) is some other function of Y, polynomial or transcedental. Then, the actual behavior of Y and t will depend on F(Y).

Related Questions

Navigate

• Browse (All)
• Browse Q
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.