Explanation Given: The given system is { d x d t = 2 x + 2 y d y d t = x + 3 y . Calculation: The given system in the matrix form, X ′ = [ 2 2 1 3 ] X Here A = [ 2 2 1 3 ] 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 | 2 − λ 2 1 3 − λ | = 0 ( 2 − λ ) ( 3 − λ ) − 2 = 0 λ 2 − 5 λ + 4 = 0 So, the value of the λ is, λ 2 − 5 λ + 4 = 0 ( λ − 1 ) ( λ − 4 ) = 0 λ = 1 , 4 Now, find the Eigen vectors for the distinct Eigen values, ( A − λ 1 I ) X = 0 ( [ 2 2 1 3 ] − 1 [ 1 0 0 1 ] ) ( x y ) = ( 0 0 ) ( [ 1 2 1 2 ] ) ( x y ) = ( 0 0 ) This system is equivalent to, x + 2 y = 0 2 y = − x y = − 1 2 x Let x = − 2 p and y = p , then X ( t ) = p ( − 2 1 )

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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