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# 0 0 spiral point asymptotically stable 1 2 1 2 saddle

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(0 , 0), spiral point, asymptotically stable; (1 2 , 1 + 2), saddle point, unstable; (1 + 2 , 1 2), saddle point, unstable 9. a., c. (0 , 0), saddle point, unstable; (2 , 2), spiral point, asymptotically stable; ( 1 , 1), spiral point, asymptotically stable; ( 2 , 0), saddle point, unstable 10. a., c. (0 , 0), saddle point, unstable; ( 2 , 2), node, unstable; (4 , 4), spiral point, asymptotically stable 11. a., c. (0 , 0), saddle point, unstable; (2 , 0), saddle point, unstable; (1 , 1), spiral point, asymptotically stable; ( 2 , 2), spiral point, asymptotically stable 12. a., c. (0 , 0), node, unstable; (1 , 1), saddle point, unstable; (3 , 1), spiral point, asymptotically stable 13. a., c. (0 , 1), saddle point, unstable; (1 , 1), node, asymptotically stable; ( 2 , 4), spiral point, unstable 14. a. 4 x 2 y 2 = c 15. a. 4 x 2 + y 2 = c 16. a. ( y 2 x ) 2 ( x + y ) = c 17. a. arctan ( y x ) − ln x 2 + y 2 = c 18. a. x 2 y 2 3 x 2 y 2 y 2 = c 19. a. y 2 2 − cos x = c 20. a. x 2 + y 2 x 4 12 = c Section 9.3, page 415 1. linear and nonlinear: saddle point, unstable 2. linear: center, stable; nonlinear: spiral point or center, indeterminate 3. linear: improper node, unstable; nonlinear: node or spiral point, unstable 4. a., b., c. (0 , 0); u = − 2 u + 2 v, v = 4 u + 4 v ; r = 1 ± 17; saddle point, unstable ( 2 , 2); u = 4 u, v = 6 u + 6 v ; r = 4 , 6; node, unstable (4 , 4); u = − 6 u + 6 v, v = − 8 u ; r = − 3 ± 39 i ; spiral point, asymptotically stable 5. a., b., c. (0 , 0); u = u, v = 3 v ; r = 1 , 3; node, unstable (1 , 0); u = − u v, v = 2 v ; r = − 1 , 2; saddle point, unstable ( 0 , 3 2 ) ; u = 1 2 u, v = 3 2 u 3 v ; r = 1 2 , 3; node, asymptotically stable ( 1 , 2); u = u + v, v = − 2 u 4 v ; r = ( 3 ± 17) 2; saddle point, unstable 6. a., b., c. (1 , 1); u = − v, v = 2 u 2 v ; r = − 1 ± i ; spiral point, asymptotically stable ( 1 , 1); u = − v, v = − 2 u 2 v ; r = − 1 ± 3; saddle point, unstable 7. a., b., c. (0 , 0); u = − 2 u + 4 v, v = 2 u + 4 v ; r = (1 ± 17 2; saddle point, unstable (2 , 1); u = − 3 u + 6 v, v = − 4 u ; r = 3 ± 87 i 4 ; spiral point, asymptotically stable (2 , 2); u = − 6 v, v = 2 u ; r = ± 3 i ; center or spiral point, indeterminate (4 , 2); u = − 8 v, v = − 2 u 4 v ; r = − 1 ± 5; saddle point, unstable 8. a., b., c. (0 , 0); u = u, v = v ; r = 1 , 1; node or spiral point, unstable ( 1 , 0); u = − u, v = 2 v ; r = − 1 , 2; saddle point, unstable 9. a., b., c. (0 , ± 2 n 𝜋 ) , n = 0 , 1 , 2 , ; u = v, v = − u ; r = ± i ; center or spiral point, indeterminate (2 , ± (2 n 1) 𝜋 ) , n = 1 , 2 , 3 , ; u = − 3 v, v = − u ; r = ± 3; saddle point, unstable 10. a., b., c. (0 , 0); u = u, v = v ; r = 1 , 1; node or spiral point, unstable (1 , 1); u = u 2 v, v = − 2 u + v ; r = 3 , 1; saddle point, unstable 11. a., b., c. (1 , 1); u = − u v, v = u 3 v ; r = − 2 , 2; node or spiral point, asymptotically stable ( 1 , 1); u = u + v, v = u 3 v ; r = − 1 ± 5; saddle point, unstable

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Boyce 9131 BMAnswersToProblems 2 March 11, 2017 15:55 595 Answers to Problems 595 12. a., b., c.
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• Spring '16
• Anhaouy
• Districts of Vienna, Boyce, e2t, 3y, = min, + c2 sin x

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