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Consider the differential equation for the vector-valued function x, x = x, A- Find the eigenvalues...
How to do Part 3? -- Find e^(At), the exponential of matrix A, where t ∈ ℝ is any real number.
How to do Part 3? -- Find e^(At), the exponential of matrix A,
where t ∈ ℝ is any real number.
Part 1: Finding Eigenpairs [10 10 5 10 -5 Find the eigenvalues λ,A2 and their corresponding eigenvectors vi , v2 of the matrix A- (a) Eigenvalues: 1,222.3 (b) Eigenvector for 21 you entered above: Vi = <-1/2,1> (c) Eigenvector for 22 you entered above: Part 2: Diagonalizability (d) Find a diagonal matrix D and an invertible matrix P D,...
Consider the differential equation y" – 7 ý + 12 y = 3 e21. (a) Find...
Consider the differential equation y" – 7 ý + 12 y = 3 e21. (a) Find r1, r2, roots of the characteristic polynomial of the equation above. W r1, r2 = 3,4 (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. yı(t) = e^(3t) M y2(t) = e^(41) (c) Find a particular solution Yp of the differential equation above. M yp(t) = Note: You can earn partial credit on this problem.
Consider the differential equation (a) Find ri, r2, roots of the characteristic polynomial of the equation...
Consider the differential equation (a) Find ri, r2, roots of the characteristic polynomial of the equation above. T1,T2 (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. n(t) = v2(t) (c) Find a particular solution yp of the differential equation above. Bplt)
Problem 5. (1 point) Consider the linear system a. Find the eigenvalues and eigenvectors for the...
Problem 5. (1 point) Consider the linear system a. Find the eigenvalues and eigenvectors for the coefficient matrix. and iz = b. Find the real-valued solution to the initial value problem - -3y - 2y2 Syı + 3y2 yı(0) = -7, (0) = 10 Use I as the independent variable in your answers. Y() = Note: You can earn partial credit on this problem. Problem 6. (1 point) Find the most general real-valued solution to the linear system of differential...
4 Matrix A is defined as A = [3_21 (a) Find the eigenvalues. (5 marks) (b)...
4 Matrix A is defined as A = [3_21 (a) Find the eigenvalues. (5 marks) (b) Find a corresponding eigenvector for each of the eigenvalues found in (a). (10 marks) (c) Use the above (a) and (b) results to solve the vector-matrix differential equation * = 1} 21x with the initial conditions X(O) = (0) (10 marks)
Section 6.1 Eigenvalues and Eigenvectors: Problem 10 Previous Problem Problem List Next Problem 4 and the...
Section 6.1 Eigenvalues and Eigenvectors: Problem 10 Previous Problem Problem List Next Problem 4 and the determinant is det(A) --- 45. Find the eigenvalues of A. (1 point) Suppose that the trace of a 2 x 2 matrix A is tr(A) smaller eigenvalue larger eigenvalue Note: You can earn partial credit on this problem Preview My Answers Submit Answers Section 6.1 Eigenvalues and Eigenvectors: Problem 8 Previous Problem Problem List Next Problem (1 point) Find the eigenvalues di < 12...
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...
Find all eigenvalues and eigenvector of the matrix 2 2 A 1 1 -2 -4-1 Give...
Find all eigenvalues and eigenvector of the matrix 2 2 A 1 1 -2 -4-1 Give the eigenvalues in ascending order. Choose the corresponding eigenvectors from the table below: 0 1 -2 2 1 V 2 = A 0 2 Vector 1 Vector 2 Vector 3 Vector 4 Vector 5 Vector 6 Eigenvector number: Eigenvector number: A3 Eigenvector number: Il
Consider the differential equation e24 y" – 4y +4y= t> 0. t2 (a) Find T1, T2,...
Consider the differential equation e24 y" – 4y +4y= t> 0. t2 (a) Find T1, T2, roots of the characteristic polynomial of the equation above. 11,12 M (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. yı(t) M y2(t) = M (C) Find the Wronskian of the fundamental solutions you found in part (b). W(t) M (d) Use the fundamental solutions you found in (b) to find functions ui and Usuch...
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...
How to do Part 3? -- Find e^(At), the exponential of matrix A,
where t ∈ ℝ is any real number.
Part 1: Finding Eigenpairs [10 10 5 10 -5 Find the eigenvalues λ,A2 and their corresponding eigenvectors vi , v2 of the matrix A- (a) Eigenvalues: 1,222.3 (b) Eigenvector for 21 you entered above: Vi = <-1/2,1> (c) Eigenvector for 22 you entered above: Part 2: Diagonalizability (d) Find a diagonal matrix D and an invertible matrix P D,...
Consider the differential equation y" – 7 ý + 12 y = 3 e21. (a) Find r1, r2, roots of the characteristic polynomial of the equation above. W r1, r2 = 3,4 (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. yı(t) = e^(3t) M y2(t) = e^(41) (c) Find a particular solution Yp of the differential equation above. M yp(t) = Note: You can earn partial credit on this problem.
Consider the differential equation (a) Find ri, r2, roots of the characteristic polynomial of the equation above. T1,T2 (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. n(t) = v2(t) (c) Find a particular solution yp of the differential equation above. Bplt)
Problem 5. (1 point) Consider the linear system a. Find the eigenvalues and eigenvectors for the coefficient matrix. and iz = b. Find the real-valued solution to the initial value problem - -3y - 2y2 Syı + 3y2 yı(0) = -7, (0) = 10 Use I as the independent variable in your answers. Y() = Note: You can earn partial credit on this problem. Problem 6. (1 point) Find the most general real-valued solution to the linear system of differential...
4 Matrix A is defined as A = [3_21 (a) Find the eigenvalues. (5 marks) (b) Find a corresponding eigenvector for each of the eigenvalues found in (a). (10 marks) (c) Use the above (a) and (b) results to solve the vector-matrix differential equation * = 1} 21x with the initial conditions X(O) = (0) (10 marks)
Section 6.1 Eigenvalues and Eigenvectors: Problem 10 Previous Problem Problem List Next Problem 4 and the determinant is det(A) --- 45. Find the eigenvalues of A. (1 point) Suppose that the trace of a 2 x 2 matrix A is tr(A) smaller eigenvalue larger eigenvalue Note: You can earn partial credit on this problem Preview My Answers Submit Answers Section 6.1 Eigenvalues and Eigenvectors: Problem 8 Previous Problem Problem List Next Problem (1 point) Find the eigenvalues di < 12...
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...
Find all eigenvalues and eigenvector of the matrix 2 2 A 1 1 -2 -4-1 Give the eigenvalues in ascending order. Choose the corresponding eigenvectors from the table below: 0 1 -2 2 1 V 2 = A 0 2 Vector 1 Vector 2 Vector 3 Vector 4 Vector 5 Vector 6 Eigenvector number: Eigenvector number: A3 Eigenvector number: Il
Consider the differential equation e24 y" – 4y +4y= t> 0. t2 (a) Find T1, T2, roots of the characteristic polynomial of the equation above. 11,12 M (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. yı(t) M y2(t) = M (C) Find the Wronskian of the fundamental solutions you found in part (b). W(t) M (d) Use the fundamental solutions you found in (b) to find functions ui and Usuch...
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...
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