Explanation Given: The given system is X ′ = ( 0 1 − 2 − 2 ) X . Where the coefficient matrix is A = ( 0 1 − 2 − 2 ) . Definition used: For any matrix, the inverse of the matrix can be obtained using adjoint matrix of A defined as, A − 1 = adj A | A | Where adj A is adjoint of matrix A and | A | is the determinant of the matrix. Procedure used: In order to find the value of e A t for the given matrix A , compute the matrix ( s I − A ) and then find ( s I − A ) − 1 followed by Inverse Laplace transformation. Formula used: The inverse Laplace transform of 1 s + a is e − a t . The inverse Laplace transform of b ( s + a ) 2 + b 2 is e − a t sin b t . The inverse Laplace transform of s + a ( s + a ) 2 + b 2 is e − a t cos b t The general solution of the system X ′ = A X is given by X = e A t C where C is constant. Calculation: The matrix is A = ( 0 1 − 2 − 2 ) . Compute the value of ( s I − A ) , ( s I − A ) = s ( 1 0 0 1 ) − ( 0 1 − 2 − 2 ) = ( s − 1 2 s + 2 ) Compute the value of ( s I − A ) − 1 by using the definition of inverse of matrix. The determinant of ( s I − A ) is computed as follows.\ | s I − A | = | s − 1 2 s + 2 | = s 2 + 2 s + 2 = ( s 2 + 2 s + 1 ) + 1 = ( s + 1 ) 2 + 1 And the value of adj ( s I − A ) is adj ( s I − A ) = ( s + 2 1 − 2 s )

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 8.4, Problem 18E

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To compute: The value of eAt using the coefficient matrix and then find the general solution of the given system.

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Chapter 8.4, Problem 17E

Chapter 8.4, Problem 19E

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A First Course in Differential Equations with Modeling Applications (MindTap Course List)

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The value of e A t using the coefficient matrix and then find the general solution of the given system.

Solution Summary: The author explains how to compute the value of eAt using the coefficient matrix and then find the general solution of the given system.

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To compute: The value of eAt using the coefficient matrix and then find the general solution of the given system.

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