Explanation Given: A matrix A = [ 0 b − b 0 ] where b > 0 . Calculation: The solution to any differential equation x ′ = A x is x ( t ) = e λ t v , where v is an eigenvector and λ is the eigenvalue corresponding to v . Consider ( A − λ I ) v = 0 where v = [ x y ] Now, the system will have non-trivial solutions if and only if det ( A − λ I ) = 0 . | − λ b − b − λ | = 0 λ 2 + b 2 = 0 λ = − b 2 λ = ± b i For λ = b i , consider ( A − ( b i ) I ) v = 0 . [ − b i b 0 − b − b i 0 ] R 1 ↔ R 2 [ − b − b i 0 − b i b 0 ] R 2 → R 2 − i ⋅ R 1 [ − b − b i 0 0 0 0 ] The equation obtained is − b x − b i y = 0 . Let y = 1 , then x = − i . This implies v = [ − i 1 ] . Now, v has complex components, this implies that corresponding solutions to x ′ = A x will also be complex conjugates. Substitute b i for λ , [ − i 1 ] for v and use the Euler’s formula e i x = cos x + i sin x in x ( t ) = e λ t v .

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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The general solution to x′=Ax and describe the behavior of the solutions.

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The general solution to x ′ = A x and describe the behavior of the solutions.

Solution Summary: The author explains the general solution to xprime=Ax and describes the behavior of the solutions.

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The general solution to x′=Ax and describe the behavior of the solutions.

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The general solution to x ′ = A x and describe the behavior of the solutions.

Solution Summary: The author explains the general solution to xprime=Ax and describes the behavior of the solutions.

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To find: The general solution to x′=Ax and describe the behavior of the solutions.

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Chapter 9 Solutions

To find:

The general solution to x′=Ax and describe the behavior of the solutions.