Problem 6 (3 points) The general solution of the system of the linear system * =...
2. The linear system 12 gives good approximations to the nonlinear system near (0, 0). (a) Sketch a phase portrait of this linear system. Identify equilibrium and straight line 1 solutions. (b) Is the equilibrium stable? (c) If zi (0) = z2(0)-1, find the smallest t > 0 such that zi (t)-0. 2. The linear system 12 gives good approximations to the nonlinear system near (0, 0). (a) Sketch a phase portrait of this linear system. Identify equilibrium and straight...
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 linear system y⃗ ′=[6−124−8]y⃗ . Problem 1. (10 points) Consider the linear system 4 ' = [-12 -8 a. Find the eigenvalues and eigenvectors for the coefficient matrix. te and 12 = v2 = b. For each eigenpair in the previous part, form a solution of y' = Ay. Use t as the independent variable in your answers. gi(t) = and yz(t) = c. Does the set of solutions you found form a fundamental set (i.e., linearly independent...
PROBLEM 3. Suppose that the general solution of a 2-by-2 system x' = Ax is x(t) = Cje-t + Czezt, y(t) = 2Cje-t – Cze2t. Sketch the phase portrait of the system and determine the matrix A.
Chapter 3, Section 3.3, Question 02 Consider the given system of equation. 2 -4 X 6 -8 (a) Find the general solution of the given system of equation 1 +c2e2t VI The general solution is given by X (t) = ci where V2. |and 21 >A2 =| ; vi = and v2 (b) Draw a direction field and a phase portrait. Describe the behavior of the solutions as t - o. 1) If the initial condition is a multiple of...
3) Write a general solution in the form Y(o)-kke to a linear system Y'-AY such that solutions living on the line y=9x head directly away fromthe origin, and solutions living on the line y =-2x head directly toward the origin. Of course, the solution at the origin does not move. Hint: You will have some freedom in picking your eigenvalues and eigenvectors, but not total freedom. yi
Please a- c for non linear system b 3. For each of the given non-linear systems, (a) find the equilibrium points, (b) near each equilibrium point, sketch the phase portrait of the linearized system, (c) use the information in (a) and (b) to sketch the phase portrait of the system: x' = - 4x + 4xy Sx = 2x – 2x² + 5xy ly=2y-y² – ry ly' = y - 2y2 + 2xy
Here is the phase portrait of a homogeneous linear system of differential tions. 4. equa- (a) Classify the equilibrium (b) If λί is the eigenvalue with corresponding eigenvector (1,1) and A2 is the eigenvalue with corresponding eigenvector (-1,3), place the three numbers 0, λ, and λ2 in order frorn least to greatest. (c) If ((t), y(t) is the solution satisfying the initial condition (x(0),y(0)- (-2,2). Find i. lim r(t) i. lim rlt) ii. lim y(t) iv. lim y(t) Here is...
Converting to linear system for three different cases: ii) y 0 For each cases provide general solutions, the phase portrait and the value of gamma at which there is a bifurcation, Converting to linear system for three different cases: ii) y 0 For each cases provide general solutions, the phase portrait and the value of gamma at which there is a bifurcation,
Could you help with this problem, please? Thank you. Compute the general solution of the system 1 dY dt Y-()* and sketch its phase portrait.