HLet A be the matrix3A₂ = -1 and corresponding eigenvectors are v₁ =V2 =HLet x(t) be the position of a particle at time t that satisfies theinitialvalue problem x' = Ax, x(0) =If the unique solution isx(t) = [22 (8)][20]=[]=C1C1C2IFm=whose eigenvalues are >₁ = -2,[3]n=Q Searchemt14+C₂2[3]ONentandthen

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Answered: Let A be the matrix A₂ = -1 and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Express the given system of higher-order differential equations as a matrix system in normal form. x"' + 3x+6y=0 y" - 5x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? O A. X₁ X₁ X₂=X", X₂ =Y₁X₂ =y" OB. X₁ X¹, X₂=X", X₂ = Y', X₂=y" OC. X₁ = x¹, x₂ = y' O D. X₁ =X, X₂=X', X3 = Y, X4 = Y' Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) O A. co Xq' X₂ m A 11 II X₁ X₂ X3 X4
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For a linear DDS with multiple equations, assume that A is the matrix of the system with eigenvalue c and eigenvector u. Which of the following is true? Check all that apply. X(n) = A"X(0) is the recursive equation for the DDS □X(n) = A"X(0) is the explicit solution for the DDS □X(n) = AX(n − 1) is the recursive equation for the DDS We can always compute X(n) as X(n) = c"X(0) using the eigenvalue c We can compute X(n) as X(n) = X(0) when X(0) is a multiple of
The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x'₁ = 4x₁ + 3x2+4x3, x'₂ = 3x₁ + 5x2 + 3x3, x'3 = 4×₁ + 3×2 + 4x3 What is the general solution in matrix form? x(t)= C₁ + 3C2e111 + C3e2t -4c2e111-2c3e21 11t -C₁-11c2e +
b) A system of differential equation is given as follows: x₁ = 9x₁ - 4x₂ x₂ = 8x₁ - 3x₂ where x₁ and X2 are functions of . Given that the eigenvalues of A= and 5 with the corresponding eigenvectors i) Write a matrix P that diagonalizes A. ii) Write a diagonal matrix D such that P¹ AP = D. iii) Hence, find the general solution to the system. [4=[83] are 1 and respectively.
The behaviour of an unforced mechanical system is governed by the differential equation 5 2 -1 0 x(1) = 3 6-9 x(t), x(0) = 1 1 1 1 0 (a) Show that the eigenvalues of the system matrix are 6, 3, 3 and that there is only one linearly independent eigenvector corresponding to the eigenvalue 3. Obtain the eigenvectors corresponding to the eigenvalues 6 and 3 and a further generalized eigenvector for the eigenvalue 3. (b) Write down a generalized modal matrix M and confirm that AM = MJ for an appropriate Jordan matrix J. x(t) = Me M-¹x(0) obtain the solution to the given differential equation. (c) Using the result
A system of ordinary differential equations has 2 x 2 state matrix with eigenvalues and corresponding eigenvectors (eigenpairs) A1 = 5, v1 = and A2 = -6, Based on the given eigenpairs write down the general solution to the associated system of differential equations. The solution must be written as a |¤1(t) single vector like a =
Consider the system of differential equations = (-3/2 3/2)x+ (-¹1) ¹ x' = -3/2 -1 where two eigenvalues of 2 3/2 -3/2 -1 are r = 1/2 and r₂ = 1/2, and the corresponding eigenvector is v= -(2/3). Let x = Ty where T is a transformation matrix. Find two first order differential equations, eigenvector is u=== and = -/+nte, n =+², ₂ -+-+ M and the generalized
Suppose that a 2 x 2 matrix A with real entries has an eigenvalue X = 1 + 3i with corresponding 4 i [12] eigenvector 1 + 2i Which of the following is NOT a solution of the linear system y' = Ay? y(t) = et y(t) = et y(t) = et y(t) = et 4 cos(3t) sin(3t) [cos(3t) + 2 sin(3t). 4 cos(3t) + sin(3t) cos(3t) 2 sin (3t) - cos(3t) + 4 sin(3t) 2 cos(3t) + sin(3t) cos (3t) - 4 sin (3t) -2 cos(3t) - sin(3t)
Need only a handwritten solution only (not a typed one).

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