Q5. (15pt] Consider the following system of differential equations. it t = = Ctyt - 1,...
Please solve the following problem, solve all parts
3. Consider the following system of autonomous differential equations for the populations of two species: dx dt dy dt --0.2y0.0004 ry 0.1 x 0.001 ry a) What type of system might this represent (and why) ? b) Are there equilibria? If yes, what are they? c) Perform a graphical analysis and sketch some trajectories in the phase plane. Comment on the stability of any equilibria. d) What would you predict for the...
Consider the nonlinear System of differential equations di dt dt (a) Determine all critical points of the system (b) For each critical point with nonnegative x value (20) i. Determine the linearised system and discuss whether it can be used to approximate the ii. For each critical point where the approximation is valid, determine the general solution of iii. Sketch by hand the phase portrait of each linearised system where the approximation behaviour of the non-linear system the linearised system...
Consider the following linear system of differential equations: dx/dt = 2x-3y dy/dt = -x +4y (a) Write this system of differential equations in matrix form (b) Find the general solution of the system (c) Solve the initial value problem given x(0) = 3 and y(0) = 4 (d) Verify the calculations with MATLAB
1. (20 marks) This question is about the system of differential equations dY (3 1 (a) Consider the case k 0 i. Determine the type of equilibrium at (0,0) (e.g., sink, spiral source). i. Write down the general solution. ili Sketch a phase portrait for the system. (b) Now consider the case k -3. (-1+iv ) i. In this case, the matrix has an eigenvalue 2+i/2 with eigenvector and an eigenvalue 2-W2 with eigenvector Determine the type of equilibrium at...
consider the system of differential equations ; 1) Find the fixed points of the system , 2) Evaluate the Jacobian Matrix at each fixed point, 3) Classify stability of each fixed point, 4) Sketch the graph of the phase portrait,
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7. (20 points) Consider the system of nonlinear equations: a) The system has 4 critical points. Find them. b) One of the critical points is (-1, -1). Linearize the system at that point. c) Based on the linear system you derived in b), classify the type and stability of point (-1, -1).
7. (20 points) Consider the system of nonlinear equations: a) The system has 4 critical points. Find them. b) One of the critical points is (-1, -1)....
2. Given the system of differential equations i = Ar, where A = 2 ) (1) Show that for any value of parameters a, b, c, with det(A) 70, the stability of the system implies the parametrization of Hyperbola and Ellipses in the phase space of the system. in this context, these types of matrices generate a linear Hamiltonian system
Question 3 Consider the following linear system of differential equations dx: = 2x-3y dt dy dt (a) Write this system of differential equations in matrix form (b) Find the general solution of the system (c) Solve the initial value problem given (0) 3 and y(0)-4 (d) Verify the calculations with MATLAB
Question 3 Consider the following linear system of differential equations dx: = 2x-3y dt dy dt (a) Write this system of differential equations in matrix form (b) Find the...
Find the most general real-valued solution to the linear system
of differential equations
(1 point) a. Find the most general real-valued solution to the linear system of differential -5 -36 x. -5 equations x 1 CHH x1 (t) = C1 x2 (t) b. In the phase plane, this system is best described as a O source/ unstable node Osink /stable node Osaddle center point ellipses Ospiral source spiral sink none of these tsi O O O
(1 point) a. Find...
8. Consider the nonhomogeneous linear system of differential equations 1 1 1 -1 u = -1 11 1 1 u-et 1 1 2 3 First of all, find a fundamental matrix and the inverse matrix of the fundamental matrix of the corresponding homogeneous linear system. Then given a particular solution 71 uy(t) = et 1 2 find the general solution of the nonhomogeneous linear system of differential equations. Hint: det(A - \I) = -(1 – 2)?(1+1)