Consider the non linear ODE: (dx/dt) = -y = f(x,y) (dy/dt) = x^2-x = g(x,y) (a)....

Consider the non linear ODE:

(dx/dt) = -y = f(x,y)

(dy/dt) = x^2-x = g(x,y)

(a). Compute all critical points (b) Derive the Jacobian matrix (c). Find the Jacobians for each critical point (d). Find the eigenvalues for each Jacobian matrix (e). Find the linearized solutions in the neighborhood of each critical point (f) Classify each critical point and discuss their stability (g) Sketch the local solution trajectories in the neighborhood of each critical point

Expert Solution

Colby Messinger answered 4 months ago

dx dt =ax+by dy dt =−x − y, 2. As the values of a and b...

dx dt =ax+by dy dt =−x − y, 2. As the values of a and b are changed so that the point (a,b) moves from one region to another, the type of the linear system changes, that is, a bifurcation occurs. Which of these bifurcations is important for the long-term behavior of solutions? Which of these bifurcations corresponds to a dramatic change in the phase plane or the x(t)and y(t)-graphs?

Consider the following second-order ODE: (d^2 y)/(dx^2 )+2 dy/dx+2y=0 from x = 0 to x =...

Consider the following second-order ODE: (d^2 y)/(dx^2 )+2 dy/dx+2y=0 from x = 0 to x = 1.6 with y(0) = -1 and dy/dx(0) = 0.2. Solve with Euler’s explicit method using h = 0.4. Plot the x-y curve according to your solution.

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dx/dt = -5x + y , dy/dt = 4x - 2y

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