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20DF06hw6sol - Math 20D Homework 6 Solutions 7.6 10 First...

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Math 20D Homework 6 Solutions November 10, 2006 7.6 10. First find e-values and e-vectors: ( - 3 - λ )( - 1 - λ ) + 2 = λ 2 + 4 λ + 5 = 0 λ = - 4 ± 16 - 20 2 = - 2 ± i Complex e-values, so just need one: λ = - 2 + i . - 1 - i 2 - 1 1 - i x y = 0 0 ( - i - 1) x + 2 y = 0 ~v = 2 1 + i Expand ~ve λt using Euler’s formula, and find real and imaginary parts: 2 1 + i e ( - 2+ i ) t = 2 1 + i e - 2 t (cos t + i sin t ) = 2 e - 2 t cos t e - 2 t cos t - e - 2 t sin t + i 2 e - 2 t sin t e - 2 t cos t + e - 2 t sin t Fundamental matrix: Ψ( t ) = 2 e - 2 t cos t 2 e - 2 t sin t e - 2 t cos t - e - 2 t sin t e - 2 t cos t + e - 2 t sin t Ψ(0) = 2 0 1 1 Ψ - 1 (0) = 1 / 2 0 - 1 / 2 1 Solution to IVP: ~x ( t ) = Ψ( t - 1 (0) ~x (0) = 2 e - 2 t cos t 2 e - 2 t sin t e - 2 t cos t - e - 2 t sin t e - 2 t cos t + e - 2 t sin t 1 / 2 0 - 1 / 2 1 1 - 2 = 2 e - 2 t cos t 2 e - 2 t sin t e - 2 t cos t - e - 2 t sin t e - 2 t cos t + e - 2 t sin t 1 / 2 - 5 / 2 = e - 2 t cos t - 5 sin t - 2 cos t - 3 sin t 12. (a) ( - 4 / 5 - λ )(6 / 5 - λ ) + 2 = λ 2 - 2 / 5 λ + 26 / 5 = 0 λ = 2 / 5 ± 4 / 25 - 104 / 25 2 = 1 / 5 ± i (b) Since the real part of λ is positive, the equilibrium point at the origin is a spiral source. To determine clockwise or counterclockwise, compute - 4 / 5 2 - 1 6 / 5 1 0 = - 4 / 5 - 1 and - 4 / 5 2 - 1 6 / 5 0 1 = 2 6 / 5 . Vectors along the x 1 -axis point down and to the left; vectors along the 1
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2 x 2 -axis point up and to the right. This implies that trajectories are spiraling clockwise away from the origin.
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