Consider the 3 first-order systems(1)x = 3x - 2yy = 4x - yx = 3x - 2yý= 2x - 2y(3)x = -2x + yy = x - 2y(a) For each system identify the coefficient matrix and its eigenvalues.(b) Using the information from (a), classify each equilibrium at o for each system accordingto its type and stability.(c) Use Sage to plot a slope field for each system.(d) For system (3), find the solution to the IVP x(0) = 3, y(0) = 1.

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Answered: Consider the 3 first-order systems (1)…

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 A be a 2 x 2 diagonalizable matrix. 3et+ ecit If is the solution to the system ở' (t) = Ax(t), 3e²t + ec₂t then check All the possible values of c₁ and c2 below. (a)c₁ = 1; (b) C₁ = 2; (c) C₁ = 3; (d) c₂ = = 1; (e) c₂ = 2; (f) C₂ (a) (b) (c) (d) (e) (f) U 1 U U = 3.
A single-input, single output system has the state variable representation a. b. C. d. where A = = L-₁0 B = 1 -10 -5. The transfer function of the system T(s) = Y(s)/U(s) is: a. T(S) c. T(S): = b. T(S) = d. T(S) = -50 5s²+5s+1 -100 s²+10s+100 -50 s²+5s+100 x = Ax + Bu y = Cx -100 s²+5s+10 =[] c C = [010]
First verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system
dx dt = x+y+z; (a) (b) dy dt = x + 3y - z; dz dt = 2y + 2z For the system above, define matrix A and vector X and write the system in matrix form. Show all work to solve the system and clearly state the general solution vector X
Transform the equation u" +4u' + 3u = 0 into a system of first-order equations.
Verify that X(t) is a fundamental matrix for the given system and compute X1(t). Then use the result that if X(t) is a fundamental matrix for the system x' = Ax, then x(t) = x(t)X(0)x is the solution to the initial value problem x' = Ax, x(0) = x0 - 24e-1 -9e-21 060 -5 x'= 10 1x, x(0)= 8e3t x(t)= 1 1 0 -4e-t 3e-21 3t 4e -2 -20e-t 3e-21 3t 4e 060 (a) If x(t) = [×₁ (t) x2(t) ×3(t)] and A = 10 1 validate the following identities and write the column vector that equals each side of the equation. 1 1 0 ×₁' = Ax₁ = x2' = Ax₂ = x3' = Ax3 (b) Next, compute the Wronskian of X(t). W[x()()()]= Since the Wronskian is never and each column of X(t) is a solution to x' = Ax, x(t) is a fundamental matrix. (c) Find x(t) = ☐ (d) x(t) = x(t)x¯1 (0)x₁ =
(b) For each augmented matrix determine how many solution the associated system of linear equations has: 1200 0120 0000 (i) 0 1 20 (ii) 0012 (iii) 1200 0012 0001 0120
Consider a second-order nonlinear system described by i1 = x – 02, i2 = -1 – x1 + (1+x²)u, = X1 - y = x1 + 2x1x2. (a) Find all q equilibrium points xe,1, Xe,2, ..., xe,g of the system and the corresponding outputs Y0,1, Yo,2, . .. , Y0,g when the input u has a nominal value uo = 1. (b) For each i E {1,2, ..., q}, linearize the system around the operating point (xe,i, u0, Yo,i).
Başlangıç iterasyonu x(0) =0 olmakla Gauss - Seeidel yöntemi kullanarak aşağıdaki lineer sistemin ikinci iterasyonunun birinci bileşenini (x, (2) yi) bulunuz. Find the first component (x, (2)) of second iteration of the Gauss-Seidel method for the following linear system, using the initial iteraion x(0) = 0. X₁-x₂-3x3 =3, X₁-2x₂-5x3=3, 4x₁-x₂-3x3=4.
Find a particular solution of the indicated linear system that satisfies the initial conditions x, (0) = 3, X2 (0) = 4, and x3 (0) = 8. - 18 - 22 - 2 1 1 - 2t 2t x' = 2 x; X1 = e - 4 X2 = - 1 4t X3 = e - 1 16 20 - 16 - 16 2 4 1 ... The particular solution is x, (t) =, xX2(t) =, . and x3(t) =
Suppose (). (C) are solutions to a 2-dimensional homogeneous linear system. (a) Find the coefficient matrix of this system. (b) Find the general solution of this system.

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