Explanation Formula used: The eigenvalues of the matrix A are roots of the equation | A − λ I | = 0 . The general solution of the linear homogeneous system X ′ = A X is as follows, X ( t ) = c 1 v 1 e λ 1 t + c 2 v 2 e λ 2 t + ⋯ + c n v n e λ n t = c 1 X 1 + c 2 X 2 + ⋯ + c n X n Here, λ 1 , λ 2 , ⋯ , λ n are the n distinct real eigenvalues of A with corresponding Eigen vectors v 1 , v 2 , ⋯ , v n . c 1 , c 2 , ⋯ , c n are constants. If the eigenvalues are complex conjugate λ 1 , 2 = α ± β i with corresponding Eigen vector v 1 , 2 = a ± b i , then X 1 = e α t ( a cos β t − b sin β t ) and X 2 = e α t ( b cos β t + a sin β t ) . Calculation: Consider the expression, X ′ = ( 1 − 10 − 7 10 ) X Solve | A − λ I | = 0 to find the eigenvalues. | A − λ I | = | ( 1 − 10 − 7 10 ) − λ ( 1 0 0 1 ) | = | ( 1 − 10 − 7 10 ) − ( λ 0 0 λ ) | = | ( 1 − λ − 10 − 7 10 − λ ) | = ( 1 − λ ) ( 10 − λ ) − 70 Further simplify as: ( 1 − λ ) ( 10 − λ ) − 70 = 10 − λ − 10 λ + λ 2 − 70 = λ 2 − 11 λ − 60 = λ 2 − 15 λ + 4 λ − 60 = ( λ − 15 ) ( λ + 4 ) Equate to zero and find λ : ( λ − 15 ) ( λ + 4 ) = 0 First factor is, ( λ − 15 ) = 0 λ = 15 Second factor is, ( λ + 4 ) = 0 λ = − 4 The Eigenvalues are λ 1 = 15 , λ 2 = − 4 . The Eigenvector corresponds to λ 1 = 15

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Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Chapter 6.4, Problem 13E

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To find: The general solution of X′=(1−10−710)X.

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Chapter 6.4, Problem 12E

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Chapter 6 Solutions

Introductory Differential Equations

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The general solution of X ′ = ( 1 − 10 − 7 10 ) X .

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To find: The general solution of X′=(1−10−710)X.

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