s251final(sp07)

s251final(sp07) - ---= ′ 2 2 2 2 c) Saddle point,...

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MATH 251 Final Exam May 10, 2007 ANSWER KEY 1. C 2. A 3. B 4. D 5. a) Node, asymptotically stable (a “nodal sink”) b) Saddle point, unstable c) Spiral point, unstable (a “spiral source”) d) Center, stable (but not asymptotically stable) e) Node, unstable (a “nodal source”) 6. a) a ( x ) and e ( x ) b) b ( x ) and c ( x ) 7. - + = 2 2 2 1 4 2 1 1 ) ( e e t t y t 8. a) Eigenvalues are 4 ) 1 2 ( 2 2 π λ - = n , n = 1, 2, 3, … b) Eigenfunctions are - = x n x X n 2 ) 1 2 ( cos ) ( , n = 1, 2, 3, … 9. a) (1,1), (1,-1), (-1,1), (-1,-1) b) x x
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Unformatted text preview: ---= ′ 2 2 2 2 c) Saddle point, unstable 10. = + ′ = = ′ =-′ + ′ ′ ) ( , ) ( , 7 5 tT T L X X X X X 11. 2 5 10 ) (-= x x v 12. a) 20 10 ) ( + = x x v b) ) 2 sin( 15 2 sin 20 10 ) , ( 2 3 24 2 / 3 x e x e x t x u t t π--+ -+ = c) 30 ) , 1 ( lim = ∞ → t u t 13. b) ) 1 2 ( 4--= n b n , n = 1, 2, 3, … c) f (-1) = 1, f (0) = 0, f (3) = 1...
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This note was uploaded on 07/23/2008 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Penn State.

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s251final(sp07) - ---= ′ 2 2 2 2 c) Saddle point,...

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