Explanation Calculation: The given differential system, X ′ = ( − 1 1 0 1 2 1 0 3 − 1 ) X The given differential system in matrix form X ′ = A X is given below, ( d x d t d y d t d z d t ) = ( − 1 1 0 1 2 1 0 3 − 1 ) ( x y z ) (1) Where, A = ( − 1 1 0 1 2 1 0 3 − 1 ) , X = ( x y z ) and X ′ = ( d x d t d y d t d z d t ) . The characteristic equation for the above matrix A is det ( A − λ I ) = 0 , det ( A − λ I ) = 0 | − 1 − λ 1 0 1 2 − λ 1 0 3 − 1 − λ | = 0 Solve the above characteristic equation, ( − 1 − λ ) [ ( 2 − λ ) ( − 1 − λ ) − 3 ] − ( 1 ) [ 1 ( − 1 − λ ) − 0 ] + 0 = 0 ( − 1 − λ ) [ − 2 + λ 2 − λ − 3 ] − ( − 1 − λ ) = 0 ( − 1 − λ ) [ λ 2 − λ − 6 ] = 0 − 1 ⋅ ( 1 + λ ) ( λ − 3 ) ( λ + 2 ) = 0 On further solving the above equation the values of the λ are − 1 , 3 , and − 2 . Find the eigenvectors for the eigenvalues by using the below equation. ( A − λ i I ) X = 0 ( − 1 − λ i 1 0 1 2 − λ i 1 0 3 − 1 − λ i ) ( x y z ) = ( 0 0 0 ) Further solving the above equation by using matrix multiplication. ( ( − 1 − λ i ) x + y x + ( 2 − λ i ) y + z 3 y + ( − 1 − λ i ) z ) = ( 0 0 0 ) (2) Substitute the eigenvalue λ i by λ 1 = − 1 in equation (2) and compare both sides of the matrix equation

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 8.2, Problem 9E

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To find: The general solution of the given system.

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Chapter 8.2, Problem 8E

Chapter 8.2, Problem 10E

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