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Express the following inhomogeneous system of first-order differential equations for r(t) and y(t) in matrix form: * = 2x + y + 3e¹, y = 4x - y. Write down, also in matrix form, the corresponding homogeneous system of equations. Find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. Hence write down the complementary function for the system of equations.

Express the following inhomogeneous system of first-order differential equations for r(t) and y(t) in matrix form: * = 2x + y + 3e¹, y = 4x - y. Write down, also in matrix form, the corresponding homogeneous system of equations. Find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. Hence write down the complementary function for the system of equations.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
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(a) Express the following inhomogeneous system of first-order differential equations for r(t) and y(t) in matrix form: * = 2x + y + 3e", y = 4x - y. e Write down, also in matrix form, the corresponding homogeneous system of equations. Pod (b) Find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. (c) Hence write down the complementary function for the system of equations. (d) Find a particular integral for the original inhomogeneous system. (e) Hence write down the general solution of the original inhomogeneous system. (f) Find the particular solution of the original inhomogeneous system with x = 3 and y=-3 when t = 0. (g) What is the long-term behaviour of this particular solution as t becomes large? Does the ratio y/x tend to a fixed number, and if so what number?
Transcribed Image Text:(a) Express the following inhomogeneous system of first-order differential equations for r(t) and y(t) in matrix form: * = 2x + y + 3e", y = 4x - y. e Write down, also in matrix form, the corresponding homogeneous system of equations. Pod (b) Find the eigenvalues of the matrix of coefficients and an eigenvector corresponding to each eigenvalue. (c) Hence write down the complementary function for the system of equations. (d) Find a particular integral for the original inhomogeneous system. (e) Hence write down the general solution of the original inhomogeneous system. (f) Find the particular solution of the original inhomogeneous system with x = 3 and y=-3 when t = 0. (g) What is the long-term behaviour of this particular solution as t becomes large? Does the ratio y/x tend to a fixed number, and if so what number?
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