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 5 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.

Consider the following first-order ODE dy/dx=x^2/y from x = 0 to x = 2.4 with y(0)...

Consider the following first-order ODE dy/dx=x^2/y from x = 0 to x = 2.4 with y(0) = 2. (a) solving with Euler’s explicit method using h = 0.6 (b) solving with midpoint method using h = 0.6 (c) solving with classical fourth-order Runge-Kutta method using h = 0.6. Plot the x-y curve according to your solution for both (a) and (b).

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

Equilibrium solutions of non linear ODEs dx/dt = cot(x) & dx/dt=sin(x) Using linear stability characterise the...

Equilibrium solutions of non linear ODEs dx/dt = cot(x) & dx/dt=sin(x) Using linear stability characterise the E.S.

Initial value problem : Differential equations: dx/dt = x + 2y dy/dt = 2x + y...

Initial value problem : Differential equations: dx/dt = x + 2y dy/dt = 2x + y Initial conditions: x(0) = 0 y(0) = 2 a) Find the solution to this initial value problem (yes, I know, the text says that the solutions are x(t)= e^3t - e^-t and y(x) = e^3t + e^-t and but I want you to derive these solutions yourself using one of the methods we studied in chapter 4) Work this part out on paper to...

Use a LaPlace transform to solve d^2x/dt^2+dx/dt+dy/dt=0 d^2y/dt^2+dy/dt-4dy/dt=0 x(0)=1,x'(0)=0 y(0)=-1,y'(0)=5

Assume x and y are functions of t. Evaluate dy/dt with 4xy-5x+6y^3=-126, with dx/dt=-18, and x=6,y=-2...

Assume x and y are functions of t. Evaluate dy/dt with 4xy-5x+6y^3=-126, with dx/dt=-18, and x=6,y=-2 A retail store estimates that weekly sales and weekly advertising costs x are related by s=50,000-30,000e^-0.0004x. The current weekly advertising costs are $2,500, and these costs are increasing at a rate of $400 per week. Find the current rate of change of sales per week. Use implicit differentiation to find y’ for the equation below and then evaluate y’ at the indicated point, (-4,4)....

Solve ODE (3x - 2y + 1)dx + (-2x + y + 2)dy = 0 with...

Solve ODE (3x - 2y + 1)dx + (-2x + y + 2)dy = 0 with the method of x =u + h and y = v + k

Find the general solution of the given system. dx dt = 6x + y dy dt...

Find the general solution of the given system. dx dt = 6x + y dy dt = −2x + 4y [x(t), y(t)]= _____________, _______________ (6c1+8c2)10sin(6t)+(6c2+8c1)10cos(6t), c1cos(6t)+c2sin(6t) ^above is the answer I got, which is incorrect.

Question