# sol3 - ACM 95b/100b Solutions for Problem Set 3 Sean Mauch...

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Unformatted text preview: ACM 95b/100b Solutions for Problem Set 3 Sean Mauch [email protected] February 3, 2002 1 Problem 1 Problem. Find the general solutions to the following homogeneous linear systems of ODEs. De- scribe the behavior of the solutions as t → ∞ . 1. x = 3 6- 1- 2 x 2. x = 0 1 1 1 0 1 1 1 0 x Solution. 1. We consider the problem, x = Ax ≡ 3 6- 1- 2 x . The matrix has the distinct eigenvalues λ 1 = 0, λ 2 = 1. The corresponding eigenvectors are x 1 =- 2 1 , x 2 =- 3 1 . The general solution of the system of differential equations is x = c 1- 2 1 + c 2- 3 1 e t . If c 2 = 0, then the solution is a constant. Otherwise, the solution is dominated by the second term for large time and both coordinates tend to infinity. 2. We consider the problem, x = Ax ≡ 0 1 1 1 0 1 1 1 0 x . The matrix has the eigenvalues λ 1 = λ 2 =- 1, λ 3 = 2. The corresponding linearly independent eigenvectors are x 1 = - 1 1 , x 2 = - 1 1 , x 3 = 1 1 1 . 1 The general solution of the system of differential equations is x = c 1 - 1 1 e- t + c 2 - 1 1 e- t + c 3 1 1 1 e 2 t . If c 3 = 0, then the each coordinate of the solution vanishes as t → ∞ . Otherwise, the solution is dominated by the third term for large time and all the coordinates tend to infinity. x ∼ c 3 1 1 1 e 2 t , as t → ∞ 2 Problem 2 Problem. Find a fundamental matrix for each of the following systems of ODEs using a similarity transformation to de-couple the system. 1. x = 5- 1 3 1 x 2. x = 1- 1 4 3 2 1 2 1- 1 x Solution. 1. We consider the problem x = Ax ≡ 5- 1 3 1 x . The matrix has the distinct eigenvalues λ 1 = 2, λ 2 = 4. The corresponding eigenvectors are x 1 = 1 3 , x 2 = 1 1 . The Jordan canonical form of A is J = 2 0 0 4 . A is diagonalized by a similarity transformation. J = S- 1 AS , S = 1 1 3 1 The solution of the differential equation is x = e A t c . x = e A t c = S e J t S- 1 c = S e J t c = 1 1 3 1 e 2 t e 4 t c x = e 2 t e 4 t 3e 2 t e 4 t c 2 2. We consider the problem x = Ax ≡ 1- 1 4 3 2 1 2 1- 1 x The matrix has the distinct eigenvalues λ 1 = 3, λ 2 = (- 1- √ 17) / 2 and λ 3 = (- 1 + √ 17) / 2. The corresponding eigenvectors are x 1 = 3 14 5 , x 2 = - 1 (5- √ 17) / 2 1 , x 3 = - 1 (5 + √ 17) / 2 1 . The Jordan canonical form of A is J = 3 0 (- 1- √ 17) / 2 (- 1 + √ 17) / 2 A is diagonalized by a similarity transformation. J = S- 1 AS , S = 3- 1- 1 14 (5- √ 17) / 2 (5 + √ 17) / 2 5 1 1 The solution of the differential equation is x = e A t c ....
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sol3 - ACM 95b/100b Solutions for Problem Set 3 Sean Mauch...

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