Let the eigenvalues and eigenvectors of matrix A4x4 be[:]What is the general solution of the differential equationX₁ -1, 01x(t) = c₁e-tx(t) = c₁e-t(t) = c₁e-tx(t) = c₁e-t10100, A₂ = 2, V₂ =+ c₂te²t+ c₂e²t+ c₂e²t+ c₂e²t0110011AMO+ c3e²t-4t+ c3e-²+ c3 e-4t+ c3e-4t, X3,4 -4±2i, v3,400=sin (4t)cos (4t)00sin(2t)-cos (2t)00-sin(2t)cos(2t)= Ax?+c4e²t+c4e-4t00— sin(2t)cos (2t)+c4e+00cos(4t)sin (4t).-4t=00Ati0cos(2t)- sin(2t)00cos(2t)sin(2t)0cos(2t)sin(2t).where i =√-1.

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Answered: Let the eigenvalues and eigenvectors of…

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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For the system of differential equations, []+[4] y 6 8 a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) A1, A2 = U₁ = = b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₂ = c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u₂ is the eigenvector associated with the larger eigenvalue A₂. Enter the eigenvectors as a matrix with an appropriate size. v(t) y' = d) Determine a general solution to the system by completing the following steps. i. Find v(t) = = [y−¹(t)f(t)dt . yp(t) = -7 y (t) ii. Find a particular solution y(t). = ii. Then a general solution for the system in the matrix form is
(b) Determine the eigenvalues and eigenvectors of the matrix Hence, write down the associated modal matrix P and diagonal spectrum matrix D, and use these to solve the following system of differential equations * = 4x-2y j =x+y
Consider the differential equation d x(t) = 4x − y dt d dt y(t) = 2x + y. Write the differential equation in matrix form. Characterise the equilibrium at the origin and solve the initial value problem where x(0) = 3, y(0) = 4.
a) Given the matrix A = (11) Determine the eigenvalues and corres- ponding eigenvectors of A. b) Determine the general solution of the system of linear differential equations: x₁ = I₁=I₂ x₂ = x₁ + x₂
1 0 2. Consider the identity matrix / = 1 a. Come up with three linearly independent eigenvectors of I. b. Orthonormalize your three eigenvectors.
Consider the second-order equation d 2y/dt 2 + p dy/dt +qy = 0 a. If p=q=1, compute the eigenvalues and eigenvectors of the coefficient matrix. What would be the long-term behavior of the solutions in this case? b. If p=0 and q=2 determine the general solution.
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 =
With two independent eigenvectors?
Suppose A is a 2 × 2 real matrix with an eigenvalue λ = 3 + 5i and corresponding eigenvector i * = (-²+1). v= Determine a fundamental set (i.e., linearly independent set) of solutions for ÿ' = Aỹ, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. (t) = 2(t) = → - →
3. Consider the system of differential equations x' (t) = Ax(t) for t > 0, x(0) = (b) where A € C²x2 and a : R+ → C². Solve (1) by using the matrix exponential, when (a) A A = (1), H₁ 3 1 (1)
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)
I already know the eigenvalues are -2+6i, and 2-6i, and the eigenvector is [6i,1] , and [-6i,1], im just struggling finding the overall solution

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