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Unformatted text preview: B U Department of Mathematics Math 202 Differential Equations Summer 2005 Final Exam This archive is a property of Bo˘gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a nonprofit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without nonprofit purpose may result in severe civil and criminal penalties. 1. Let A = 5 1 2 2 . Find the general solution of the system→ x = A→ x without using Laplace transforms and show that as t → ∞ ,→ x →→ 0 . Solution: Let→ x =→ ξ e rt so that A→ ξ = r→ ξ . p ( r ) = det ( A rI ) = 5 r 1 2 2 r = r 2 + 7 r + 12 = 0. p ( r ) = ( r + 3)( r + 4) = 0 ⇒ r 1 = 3 , r 2 = 4 Since we have two distinct eigenvalues we shall have two linearly independent eigen vectors. ( A r 1 I )→ ξ (1) = ( A + 3 I )→ ξ (1) = 0, ( A r 2 I )→ ξ (2) = ( A + 4 I )→ ξ (2) =→ 0 .→ ξ (1) = a 1 a 2 ,→ ξ (2) = b 1 b 2 , 2 1 2 1 a 1 a 2 = , 1 1 2 2 b 1 b 2 = ⇒ a 2 = 2 a 1 , b 2 = b 1 . Choose a 1 = 1, b 1 = 1. Then→ x (1) =→ ξ (1) e r 1 t = 1 2 e 3 t ,→ x (2) =→ ξ (2) e r 2 t = 1 1 e 4 t is a fundamental set of solutions. General solution→ x ( t ) = c 1→ x (1) + c 2→ x (2) = c 1 e 3 t 1 2 + c 2 e 4 t 1 1 ....
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This note was uploaded on 11/16/2011 for the course MATH 251 taught by Professor Gurel during the Winter '11 term at Boğaziçi University.
 Winter '11
 gurel
 Equations

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