Explanation Given: The given system is, d y d t = 3 x − y d y d t = 9 x − 3 y Calculation: For the given system, the coefficient matrix is A = ( 3 − 1 9 − 3 ) . The characteristic equation is | ( A − λ I ) | = 0 where, I is the identity matrix. Substitute the values of matrix A and I in above equation. | 3 − λ − 1 9 − 3 − λ | = 0 ( λ 2 − 9 ) + 9 = 0 λ 2 = 0 Therefore, the eigenvalues of the above system is λ = 0 with multiplicity 2 . Obtain the eigenvector K 1 corresponding to the eigenvalue λ = 0 by solving the equation ( A − λ I ) K 1 = 0 where K 1 is given by ( x 1 y 1 ) . Substitute the values of A , λ and K 1 in the equation ( A − λ I ) K 1 = 0 , [ ( 3 − 1 9 − 3 ) − ( 0 0 0 0 ) ] ⋅ ( x 1 y 1 ) = ( 0 0 ) ( 3 − 1 9 − 3 ) ⋅ ( x 1 y 1 ) = ( 0 0 ) ( 3 x 1 − y 1 9 x 1 − 3 y 1 ) = ( 0 0 ) Solve above system of equations and obtain y 1 = 3 x 1 . Assume that x 1 = p 1 for some scalar p 1 which implies y 1 = 3 p 1 . Thus, the value of ( x 1 y 1 ) is, ( x 1 y 1 ) = ( p 1 3 p 1 ) = p 1 ( 1 3 ) Therefore, the eigenvector K 1 corresponding to λ = 0 is ( 1 3 ) . Obtain the second eigenvector K 2 by solving the equation ( A − λ I ) K 2 = K 1 where, K 2 is given by ( x 2 y 2 )

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 10.2, Problem 21E

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Chapter 10.2, Problem 20E

Chapter 10.2, Problem 22E

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Solution Summary: The author explains how to find the general solution of the given system. The coefficient matrix is A=(cc3& -1 9&

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