Explanation Given: The given system is { d x d t = − 5 2 x + 2 y d y d t = 3 4 x − 2 y . Calculation: The given system in the matrix form is, X ′ = [ − 5 2 2 3 4 − 2 ] X Here A = [ − 5 2 2 3 4 − 2 ] The characteristic equation for the given matrix is det ( A − λ I ) = 0 , here A is the given system of matrix and I the identity matrix. Therefore the characteristic equation is, det ( A − λ I ) = 0 | − 5 2 − λ 2 3 4 − 2 − λ | = 0 ( − 5 2 − λ ) ( − 2 − λ ) − 3 2 = 0 ( λ + 1 ) ( 2 λ + 7 ) = 0 So, the value of the λ is ( λ + 1 ) ( 2 λ + 7 ) = 0 λ = − 1 , − 7 2 Now, find the Eigen vectors for the distinct Eigen values, ( A − λ 1 I ) X = 0 ( [ − 5 2 2 3 4 − 2 ] + 1 [ 1 0 0 1 ] ) ( x y ) = ( 0 0 ) ( [ − 3 2 2 3 4 − 1 ] ) ( x y ) = ( 0

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 10.2, Problem 4E

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: The general solution of the given system.

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