Explanation Given: A matrix A = [ a b − b a ] where a > 0 and b > 0 . Approach: Let λ = a + i b , b ≠ 0 , be an eigenvalue of the n × n (real) matrix with corresponding eigenvector v = r + i s then the two real-solutions to x ′ = A x are x 1 ( t ) = e a t [ cos b t r − sin b t s ] and x 2 ( t ) = e a t [ sin b t r + cos b t s ] . If all eigenvalues have negative real part then every solution to x ′ = A x satisfies lim t → ∞ x ( t ) = 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 . | a − λ b − b a − λ | = 0 ( λ − a ) 2 + b 2 = 0 ( λ − a ) 2 = − b 2 λ = a ± i b As the eigenvalues of A are complex, the components of the corresponding eigenvectors will be oscillatory

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Differential Equations and Linear Algebra (4th Edition)

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