20DF06hw6sol - Math 20D Homework 6 Solutions November 10,...

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Unformatted text preview: Math 20D Homework 6 Solutions November 10, 2006 7.6 10. First find e-values and e-vectors: (- 3- )(- 1- ) + 2 = 2 + 4 + 5 = 0 =- 4 16- 20 2 =- 2 i Complex e-values, so just need one: =- 2 + i .- 1- i 2- 1 1- i x y = (- i- 1) x + 2 y = 0 ~v = 2 1 + i Expand ~ve t using Eulers 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: ( 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 (0) = 2 0 1 1 - 1 (0) = 1 / 2- 1 / 2 1 Solution to IVP: ~x ( t ) = ( 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- 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- 5sin t- 2cos t- 3sin t 12. (a) (- 4 / 5- )(6 / 5- ) + 2 = 2- 2 / 5 + 26 / 5 = 0 = 2 / 5 4 / 25- 104 / 25 2 = 1 / 5 i (b) Since the real part of 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 =- 4 / 5- 1 and- 4 / 5 2- 1 6 / 5 1 = 2 6 / 5 . Vectors along the x 1-axis point down and to the left; vectors along the 1 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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20DF06hw6sol - Math 20D Homework 6 Solutions November 10,...

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