Suppose a dynamical system is modeled with a difference equation x- Ax-1 k0.1.2.3.... where 4 2 8...
Find the general solution to the system of linear differential equations X'=AX. The independent variable is t. The eigenvalues and the corresponding eigenvectors are provided for you. x1' = 12x1 - 8x2 x2 = -4X1 + 8x2 The eigenvalues are 11 = 16 and 12 = 4 . The corresponding eigenvectors are: K1 = K2= Step 1. Find the nonsingular matrix P that diagonalizes A, and find the diagonal matrix D: p = 11 Step 2. Find the general solution...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
(1 point) For the linear system c(t1 61 X' = AX, with X(t) = A = and X(0) = g(t) (6 -6 - 4 (a) Find the eigenvalues and eigenvectors for the coefficient matrix. L X1 = , X1= * , and 12 = - ,X - = (b) Write the solution of the initial-value problem in terms of X(t), y(t) x(t) = g(t) =
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and corresponding eigenvectors vand v- 1? a. is a 2x2 matrix with repeated eigenvalue λ = 0 with defect 1 (has only one linearly independent eigenvector, not two.) and corresponding eigenvector vi- 13 (5 pts) Which of the direction fields corresponds to the system x -Cx, where...
6. The vectors x-[)and X - [-] are solutions vectors corresponding to the system of differential equation X = AX (a) Use the Wronskian to show that X, and X, are linearly independent. (b) Write down a general solution to the system of equations. (e) Find the solution to the system subject to the initial condition X(0) -
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. 4 2 11 = -2, V1 = 2 0 3 12 = -3, V2= 13 = -3, V3 = 1 7 2 i) Identify three solutions to the system, xi(t), xz(t), and x3(t). ii) Use a determinant to identify values of t, if any, where X1, X2, and x3 form a fundamental set of...
we assume that X solves the differential equation X'=AX. In this problem, we will investigate the strategy to deal with repeated eigenvalues. Conside:r A=17-2-6 1. This matrix has only one eigenvalue Ao of multiplicity 3. Find the characteristic equation, the eigenvalue λ0 and an eigenvector P for λ0 2. Find vectors K. L such that (A-X0IK-P and (A-X01)L-K. Compute the matrix M-1AM where M-(PIKL) 3. Let Y -M'X. Solve the equation for Y in the following manner : first, solve...
Find the general solution of the system x' = Ax where A is the given matrix. If an initial condition is given, also find the solution that satisfies the condition. 1.1 5 2 :| -2 1 )
Write each statement as True or False (a) If an (nx n) matrix A is not invertible then the linear system Ax-O hns infinitely many b) If the number of equations in a linear system exceeds the number of unknowns then the system 10p solutions must be inconsistent ) If each equation in a consistent system is multiplied through by a constant c then all solutions to the new system can be obtained by multiplying the solutions to the original...
(1) For the following system of ODES: (i) First, convert the system into a matrix equation, then, (ii) Find the eigenvalues, 11 and 12, then, (iii) Find the corresponding eigenvectors, x(1) and x(2), and finally, (iv) Give the general solution (in vector form), ygen, of the system. (Parts (i)-(iii) will be in your work) s y = -241 + 742 y2 = yı + 4y2 General Solution: