(a) We have the matrix A as
The eigenvalues of A are simply the roots of its characteristic equation, which is
Hence the roots of the characteristic equation are
Thus, the matrix has a single eigenvalue, 2, with multiplicity 2.
Now, we have to find the eigenvectors corresponding to this eigenvalue, which are given by
Now, equating the components together, we get the same equation :-
Thus, the general form of the eigenvector is given by
this completes part (a)
(b)
For repeated eigenvalues, the general solution of the system is given by
where
and is a solution of the equation
So,
Let rho = (p1,p2) so we have
So, one possible solution is obtained by putting p2 = 0, so
Thus, we have the general solution of the system as
Simplifying we get
which is our required solution.
(c) and the phase portrait of the system is
where y is along the vertical axis and x is along the horizontal axis
Consider the linear system of first order differential equations x' = Ax, where x = x(t), t > 0, and A has the eigenvalues and eigenvectors below. Sketch the phase portrait. Please label your axes. 11 = 5, V1 = 12 = 2, V2 = ()
8. 20 pts.] Suppose that a 2 x2 matrix A has the following eigenvalues and eigenvectors: () 12, 1 r2=1, 2 2 (a) Classify the equilibrium 0 (node, saddle, spiral, center). Is it stable or unstable? (b) Sketch the trajectories of the system A , where a the phase plane below. (c) On the next page, sketch the graphs of r1 (t) and 2(t) versus t for the solution that satisfies the initial condition x1(0) = 1, x2(0) = 1...
5.4 Equilibrium Solutions and Phase Portraits 1. 2 3 3 2 . (a) Draw direction field. Use the points: (0,0), (+1,0), (0, +1), (+1, +1). (b) Draw the phase portrait. (c) Classify the equilibrium solution with its stability. 11 and 2. Suppose 2 x 2 matrix A has eigenvalues – 3 and -1 with eigenvectors respectively. (a) Find the general solution of 7' = A. (b) Draw the phase portrait. (C) Classify the equilibrium solution with its stability. 3. Suppose...
vector x' = [ the first row is 2 and 8, the second row is -1 and -2] vector x (i) Compute the eigenvalues and eigenvectors of the system. (ii) Use the eigenvalues to classify the equilibrium type of the origin. (iii) Use the eigenvectors as guides to plot a phase portrait of the system. (iv) Present a general solution to the system of ODE. (v) Find the particular solution to this system of ODE if vector x(0) = [...
1. (20 marks) This question is about the system of differential equations Y. dt=(k 1 (a) Consider the case k = 0. i. Determine the type of equilibrium at (0,0) (e.g., sink, spiral source). ii. Write down the general solution. iii. Sketch a phase portrait for the system. (b) Now consider the case k3 In this case, the matrix has an eigenvalue 2+V/2 with eigenvector i. -1+iv2 and an eigenvalue 2 iv2 with eigenvector . Determine the type of equilibrium...
Consider the homogeneous system 0 ' = Až = a) T, a ER -4 a (a) Determine the eigenvalues of A in terms of a. (if any) where the qualitative nature of phase portrait for (b) Identify the values of a system changes. (c) For each value of a that you listed in part (b), sketch the phase portrait for a value of a that is slightly less than the value(s) that you identified. For example, if you identified only...
Linear Algebra Question Suppose a dynamical system is modeled with a difference equation x- Ax-1 k0.1.2.3.... where 4 2 8 .2 1 1 7 Three linearly independent eigenvectors of A are The initial condition is x: In the manner discussed in class, express the initial condition as a linear combination of the eigenvectors of A. Then construct the general solution to the difference equation using the eigenvalues and eigenvectors of A. Finally determine the long term steady-state solution to the...
7. 20 pts. Find the general solution of the following 2 x 2 system, and express your answer in terms of real-valued functions: 1 1 (t) (t) -1 0 Solution 7. 20 pts. Find the general solution of the following 2 x 2 system, and express your answer in terms of real-valued functions: 1 1 (t) (t) -1 0 Solution
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.
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...