Explanation Given: The given system is { d x d t = x + 2 y d y d t = 4 x + 3 y . Calculation: The given system in the matrix form is, X ′ = [ 1 2 4 3 ] X Here A = [ 1 2 4 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 | 1 − λ 2 4 3 − λ | = 0 ( 1 − λ ) ( 3 − λ ) − 8 = 0 λ 2 − 4 λ − 5 = 0 So, the value of the λ is, λ 2 − 4 λ − 5 = 0 ( λ + 1 ) ( λ − 5 ) = 0 λ = − 1 , 5 Now, find the Eigen vectors for the distinct Eigen values. ( A − λ 1 I ) X = 0 ( [ 1 2 4 3 ] − 5 [ 1 0 0 1 ] ) ( x y ) = ( 0 0 ) ( [ − 4 2 4 − 2 ] ) ( x y ) = ( 0 0 ) This system is equivalent to

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 10.2, Problem 1E

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The general solution of the given system.

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