quiz 6 (10).pdf

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Department
Mathematics & Engineering Courses
Course Code
MTHE 225
Professor
Gregory G Smith

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Quiz 6a Solutions We are asked to use the eigenvalue method to find a solution to the following linear system, satisfying the specified initial conditions. ′ ′ x 1 9x 1 5x ,2 x2= −6x 1 2x 2 x1(0) = 1, x2(0) = 0 Write the system as ! ! ! ′ x1 9 5 x1 x′ = −6 −2 x 2 2 ! 9 5 The matrix A = has eigenvalues λ = 3, λ = 4, as is seen from the characteristic −6 −2 polynomial: ! 9 − λ 5 det(A − λI) = det −6 −2 − λ = (9 − λ)(−2 − λ) + 30 = λ − 7λ + 12 = (λ − 3)(λ − 4) To find the eigenvector v corresponding to λ = 3, we solve the system ! ! ! 9 − 3 5 v1 = 0 −6 −2 − 3 v 0 2 ! 5 to find v = . To find the eigenvector v corresponding to λ = 4, we solve the system −6 ! ! ! 9 − 4 5 v1 0 = −6 −2 − 4 v2 0 ! 1 to find v = . Thus the two independent solutions are
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