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problem4_sol - EE221A Problem Set 4 Solutions Fall 2011...

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EE221A Problem Set 4 Solutions - Fall 2011 Problem 1. Existence and uniqueness of solutions to differential equations. Call the first system f ( x, t ) = ˙ x 1 ˙ x 2 T and the second one g ( x ) = ˙ x 1 ˙ x 2 T . a) Construct the Jacobians: D 1 f ( x, t ) = - 1 - e t sin ( x 1 - x 2 ) e t sin( x 1 - x 2 ) 15 cos ( x 1 - x 2 ) - 1 - 15 cos( x 1 - x 2 ) , Dg ( x ) = - 1 + x 2 x 1 0 - 1 . D 1 f ( x, t ) is bounded x , and f ( x, t ) is continuous in x , so f ( x ) is globally Lipschitz continuous. But while g ( x ) is continuous, Dg ( x ) is unbounded (consider the 1,1 entry as x 2 → ∞ or the 1,2 entry as x 1 → ∞ ) so the function is not globally LC. b) Agree. Note that x 2 does not depend on x 1 ; it satisfies the conditions of the Fundamental Theorem, and one can directly find the (unique by the FT) solution x 2 ( t ) = x 2 (0) e - t 0 as t → ∞ . This solution for x 2 can be substituted into the first equation to get ˙ x 1 = - x 1 + x 1 x 2 (0) e - t = x 1 ( x 2 (0) e - t - 1 ) , which again satisfies the conditions of the Fundamental Theorem, and can be solved to find the unique solution x 1 ( t ) = x 1 (0) exp (( 1 - e - t ) x 2 (0) - t )
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