Consider a system of two linear first-order ordinary differential equations: y₁ = y₁ − Y2,a) The corresponding eigenvalues areOλ₁ = 1, λ₂ = −1b) The corresponding eigenvectors of this linear ODE system are:OI and IVOll and IIIOllI and IVwhereI:u₁ =II:u₂ =III:u₁IV:u₂1+i2(¹3²)Oλ₁ = i, λ₂ = −i= ( 2 (1 + i))• (20²+0)=2(1 + i)c) The phase portrait for this system of ODEs isOCentre OUnstable focus with spiral outOλ₁=1+i, λ₂ = 1-iStable nodey₂ = 2y₁ - y2.OI and IIOStable focus with spiral in

Given system of differential equationsy˙1=y1−y2y˙2=2y1−y2In Matrix form [y˙1y˙2]=[1−12−1][y1y2]Let…

Answered: Consider a system of two linear…

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, 3 23 84 6 - 22 4et a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) A1, A2 b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₁ = 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 λ₂. Enter the eigenvectors as a matrix with an appropriate size. y' = 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 . ii. Find a particular solution y(t). H yp(t)= y(t) ii. Then a general solution for the system in the matrix form is -8:33:
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.
Consider the following system of first order homogeneous equations -1 2 E. 0. Find the eigenvalues of the coefficient matrix A. a n= 1, r, = 2 T1=1, b. 1=-1, Oc n= 1, =2 C. T2=-2 O d. Ti=1, T= 2 72= Previous page
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 =
The eigenvalues of the system x' = x y =x are a) {i,-i} b) {0,1} c) {-1,0} d) {-1, 1}
Chads baseball team is practicing hitting in the batting cages for pitching machines pitch baseballs at the rate of 35 pitches in 5 minutes chad practiced hitting for 8 minutes how many baseballs were pitched to chad in that time
a). consider the system of differential equations. 21 5²²lt) = [ ²1 ] ²lt), where Rit) = [kilt)] 23 Find the general solution to this system, and write out the solutions for Xilt) and X₂lt), without using vector notation.
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)
If A is an (n x n)-matrix of real constants that has a complex eigenvalue and eigenvector v, then the real and imaginary parts of w(t) = elty are linearly independent real-valued solutions of x' =Ax: x₁(t) = Re(w(t)) and X₂(t) = Im(w(t)), The matrix in the following system has complex eigenvalues; use the above theorem to find the general (real-valued) solution. 0 30 -3 0 0 X 005 x(t) = = x' = Find the particular solution given the initial conditions. x(t) = = x(0) = 123
Xn Let vn be the solution to the discrete dynamical system: Yn 6xn – 13yn Oxn – 7yn Xn+1 Yn+1 [:] -3 If vo find vn- -1 Put the eigenvalues in ascending order when you enter xn, Yn below. Xn = Yn = )"+ )" ||
1b
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