2. Use Eigenvalue method to find the general solution of the given systemdxdx5x + 2y3x + 2y + 4zdta.dtb.dy2x + 2zdtdy3- 2y== r-dt4dz4x + 2y + 3zdt 3. Use the Laplace transform to solve the given differential equation.x' - 2x - 4y = 4,y' + 2x + 2y= t,x(0) = 3, y(0) = 1.%3D

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Answered: 2. Use Eigenvalue method to find the…

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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Let f(x) = 2x - 3 where 0 ≤ x ≤ 1. Expand f(x) in terms of the eigenfunctions of the Sturm-Liouville system y" + y = 0, 0
a) Write the initial value problem in vector form. b) Solve by using eigenvalues/eigenvectors. c) Write the solution in x(t), y(t) form.
a) Write the initial value problem in vector form. b) Solve by using eigenvalues/eigenvectors. c) Write the solution in x(t), y(t) form.
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 =
12- The system x = x + 4y, y' = y + x has two eigenvectors....
5. Solve the linear systems using eigenvalues and eigenvectors. dx = x+4z dt dy dt -= 2y dz -= 3x+y-3z dt
lloa [Ur ses SEC::116 Question 9 Find the general solution to the system x' = Ax where A is the given matrix. -2 -1 1 A= -1 -9 3 -1 -2 -4 Hint: -6 is an eigenvalue.
determine eigenvalues and eigenvectors if the eigenvalues are real. Also, determine the general solution of the following linear systems of differential equations:dx/dt = ax + by,dy/dt = cx + dy.
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)
and Vz - 2 arp eigenved ors pf motrx A cor fespontng to ergenvalno d, = -5 and do =5 A (vi + Vz ) = to A (sT; ) = ?

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2. Use Eigenvalue method to find the general solution of the given system dx a. dx dt dy dt 3 214 5 =-X- + 2y 2y b. 11 | | | dt dt dt 3x + 2y + 4z = 2x + 2z = 4x + 2y + 3z