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![ज)- 10 [] Re24 C-3 (+ [ 1 2 3 - Fe 2 Res S3Ct) - । 2 s 26 ल) १६ 4 र 362 test १ - -* (AE (E) - teहया) =-e* [raelime +0 - Nuet](http://img.homeworklib.com/questions/6758d1a0-10dd-11eb-9283-21a3c04477bd.png?x-oss-process=image/resize,w_560)
![The general solution is who c2CN***(470 *100= 0-9 []+(21-40) 0 --- zataG+29 Cg 0-0 3 C1+7C2+26 5) 20+46+213-0, G=l, 3C476726](http://img.homeworklib.com/questions/67f82030-10dd-11eb-9b8e-59166b01641c.png?x-oss-process=image/resize,w_560)
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Consider the linear system of first order differential equations x' = Ax, where x= x(t), t...
Consider the linear system of first order differential equations x' = Ax, where x = x(t),...
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 = ()
Find the general solution to the system of linear differential equations X'=AX. The independent variable is...
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...
02. (8,2, 5) You are provided with a system of linear equations Ax - ye, where A R r ER2,yER2 and e e Ri. Let the s...
02. (8,2, 5) You are provided with a system of linear equations Ax - ye, where A R r ER2,yER2 and e e Ri. Let the spectral decomposition of A is given by V2/2 V2/2( containing 1 0 containing eigenvalues of A and V2/22/2 Lo 5 corresponding orthonormal eigenvectors a) Determine the best approximation of the unknow vector x, when the observerd vector y 181
02. (8,2, 5) You are provided with a system of linear equations Ax - ye,...
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 ide...
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) Consider the linear system "(-1: 1) y. a. Find the eigenvalues and eigenvectors for...
(1 point) Consider the linear system "(-1: 1) y. a. Find the eigenvalues and eigenvectors for the coefficient matrix. 1 v1 = and 2 V2 b. For each eigenpair in the previous part, form a solution of y' = Ay. Use t as the independent variable in your answers. (t) = and yz(t) c. Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions? Choose
(1 point) For the linear system c(t1 61 X' = AX, with X(t) = A =...
(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) =
2 2 0 0 3" (12%) Solve the linear system x"(t) AX(t) with A 0 0...
2 2 0 0 3" (12%) Solve the linear system x"(t) AX(t) with A 0 0 4 4 a) (406) Write down the characteristic equation of the coefficient matrix λ and solve its eigenvalues, . b) (8%) Find the four independent solutions of the system.
True or False Ivp questions a) An IVP of the for y' + p(t)y = g(t),...
True or False Ivp questions
a) An IVP of the for y' + p(t)y = g(t), y(0) = yo, with p and g continuous functions defined for all tER, always has a unique differentiable solution y(t) defined for all t E R. b) To find the solution of y' + p(t)y = gi(t) + 92(t), y(0) = yo it suffices to solve y' + p(t)y = gi(t), y(0) = 0 and y' + p(t)y = 92(t), y(0) = 1 and...
4. The origin (0,0) is a critical point of the first order autonomous system x'(t)- Ax(t)...
4. The origin (0,0) is a critical point of the first order autonomous system x'(t)- Ax(t) The origin can classified as asymptotically stable if Re(A) < 0 and stable if Re(A)0 for all eigenvalues λ of A. The origin is unstable if there exists an eigenvalue λ of A where Re(A) >0. For the following systems, classify the origin 1 -3x(C) b, x'(t)=11-3 1-3x(t)
Consider the linear system X' = AX where A is defined by , where a and...
Consider the linear system X' = AX where A is defined by
,
where a and b are real numbers. Assume that the determinant of A
is not zero.
Classify the equilibrium solution (0,0) depending on the signs
of a and b. Also, sketch a few trajectories for each case,
including a few tangent vectors.
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 = ()
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...
02. (8,2, 5) You are provided with a system of linear equations Ax - ye, where A R r ER2,yER2 and e e Ri. Let the spectral decomposition of A is given by V2/2 V2/2( containing 1 0 containing eigenvalues of A and V2/22/2 Lo 5 corresponding orthonormal eigenvectors a) Determine the best approximation of the unknow vector x, when the observerd vector y 181
02. (8,2, 5) You are provided with a system of linear equations Ax - ye,...
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) Consider the linear system "(-1: 1) y. a. Find the eigenvalues and eigenvectors for the coefficient matrix. 1 v1 = and 2 V2 b. For each eigenpair in the previous part, form a solution of y' = Ay. Use t as the independent variable in your answers. (t) = and yz(t) c. Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions? Choose
(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) =
2 2 0 0 3" (12%) Solve the linear system x"(t) AX(t) with A 0 0 4 4 a) (406) Write down the characteristic equation of the coefficient matrix λ and solve its eigenvalues, . b) (8%) Find the four independent solutions of the system.
True or False Ivp questions
a) An IVP of the for y' + p(t)y = g(t), y(0) = yo, with p and g continuous functions defined for all tER, always has a unique differentiable solution y(t) defined for all t E R. b) To find the solution of y' + p(t)y = gi(t) + 92(t), y(0) = yo it suffices to solve y' + p(t)y = gi(t), y(0) = 0 and y' + p(t)y = 92(t), y(0) = 1 and...
4. The origin (0,0) is a critical point of the first order autonomous system x'(t)- Ax(t) The origin can classified as asymptotically stable if Re(A) < 0 and stable if Re(A)0 for all eigenvalues λ of A. The origin is unstable if there exists an eigenvalue λ of A where Re(A) >0. For the following systems, classify the origin 1 -3x(C) b, x'(t)=11-3 1-3x(t)
Consider the linear system X' = AX where A is defined by
,
where a and b are real numbers. Assume that the determinant of A
is not zero.
Classify the equilibrium solution (0,0) depending on the signs
of a and b. Also, sketch a few trajectories for each case,
including a few tangent vectors.
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