Explanation Given: The differential system is { d x d t = x + y d y d t = − 2 x − y . Calculation: The matrix notation of differential system { d x d t = x + y d y d t = − 2 x − y is as follows: X ′ = ( 1 1 − 2 − 1 ) X ...... ( 1 ) Here, the matrix are defined as A = ( 1 1 − 2 − 1 ) , X = ( x y ) and X ′ = ( d x d t d y d t ) . The characteristic equation for the above matrix A is det ( A − λ I ) = 0 . det ( A − λ I ) = 0 | 1 − λ 1 − 2 − 1 − λ | = 0 ( 1 − λ ) ( − 1 − λ ) + 2 = 0 λ 2 + 1 = 0 Solve the above characteristic equation. λ 2 = − 1 λ = ± i ( Square root ) Therefore, the Eigen values are complex conjugates such that λ 1 = i and λ 2 = λ 1 ¯ . Now, to obtain the Eigenvectors for the Eigenvalues substitute the value of Eigen values in the equation given below. ( A − λ i I ) X = 0 ( 1 − λ i 1 − 2 − 1 − λ i ) ( x y ) = ( 0 0 ) ...... ( 2 ) Substitute Eigen value λ i by λ 1 = i in equation ( 2 ) and solve as follows: ( 1 − i 1 − 2 − 1 − i ) ( x y ) = ( 0 0 ) Write the augmented matrix of the above equation and solve as follows: ( A − i I 0 ) = ( 1 − i 1 0 − 2 − 1 − i 0 ) = ( 1 1 1 − i 0 − 2 − 1 − i 0 ) ( R 1 → R 1 ( 1 1 − i ) ) = ( 1 1 + i 2 0 − 2 − 1 − i 0 ) = ( 1 1 + i 2 0 0 0 0 ) ( R 2 → R 2 + 2 R 1 ) Further, solve the above augmented matrix in linear equations

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 10.2, Problem 36E

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The general solution of the differential system {dxdt=x+ydydt=−2x−y.

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Chapter 10.2, Problem 35E

Chapter 10.2, Problem 37E

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