s251final(sp07)

# s251final(sp07) - -= ′ 2 2 2 2 c Saddle point unstable 10...

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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 ) ( π , n = 1, 2, 3, … 9. a) (1,1), (1,-1), (-1,1), (-1,-1) b)

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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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