# 0 0 u 2 u v v u v r 3 3 i 2 spiral point

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(0 , 0); u = − 2 u v, v = u v ; r = ( 3 ± 3 i ) 2; spiral point, asymptotically stable ( 0 . 33076 , 1 . 0924) and (0 . 33076 , 1 . 0924); u = − 3 . 5216 u 0 . 27735 v, v = 0 . 27735 u + 2 . 6895 v ; r = − 3 . 5092 , 2 . 6771; saddle point, unstable 13. a., b., c. (0 , 0); u = u + v, v = − u + v ; r = 1 ± i ; spiral point, unstable 14. a., b., c. (2 , 2); u = − 4 v, v = − 7 2 u + 7 2 v ; r = (7 ± 273) 4; saddle point, unstable ( 2 , 2); u = 4 v, v = 1 2 u 1 2 v ; r = ( 1 ± 33) 4; saddle point, unstable ( 3 2 , 2 ) ; u = − 4 v, v = 7 2 u ; r = ± 14 i ; center or spiral point, indeterminate ( 3 2 , 2 ) ; u = 4 v, v = − 1 2 u ; r = ± 2 i ; center or spiral point, indeterminate 15. a., b., c. (0 , 0); u = 2 u v, v = 2 u 4 v ; r = − 1 ± 7; saddle point, unstable (2 , 1); u = − 3 v, v = 4 u 8 v ; r = − 2 , 6; node, asymptotically stable ( 2 , 1); u = 5 v, v = − 4 u ; r = ± 2 5 i ; center or spiral point, indeterminate ( 2 , 4); u = 10 u 5 v, v = 6 u ; r = 5 ± 5 i ; spiral point, unstable 18. b., c. Refer to Table 9.3.1. 20. a. R = A, T 3 . 17 b. R = A, T 3 . 20 , 3 . 35 , 3 . 63 , 4 . 17 c. T 𝜋 as A 0 d. A = 𝜋 21. a. v c 4 . 00 22. a. v c 4 . 51 27. a. dx dt = y, dy dt = − g ( x ) c ( x ) y b. The linear system is dx dt = y, dy dt = − g (0) x c (0) y . c. The eigenvalues satisfy r 2 + c (0) r + g (0) = 0. Section 9.4, page 426 1. b., c. (0 , 0); u = 3 2 u, v = 2 v ; r = 3 2 , 2; node, unstable (0 , 2); u = 1 2 u, v = − 3 2 u 2 v ; r = 1 2 , 2; saddle point, unstable ( 3 2 , 0 ) ; u = − 3 2 u 3 4 v, v = 7 8 v ; r = − 3 2 , 7 8 ; saddle point, unstable ( 4 5 , 7 5 ) ; u = − 4 5 u 2 5 v, v = − 21 20 u 7 5 v ; r = ( 22 ± 204) 20; node, asymptotically stable 2. b., c. (0 , 0); u = 3 2 u, v = 2 v ; r = 3 2 , 2; node, unstable (0 , 4); u = − 1 2 u, v = − 6 u 2 v ; r = − 1 2 , 2; node, asymptotically stable ( 3 2 , 0 ) ; u = 3 2 u 3 4 v, v = 1 4 v ; r = 1 4 , 3 2 ; node, asymptotically stable (1 , 1); u = − u 1 2 v, v = − 3 2 u 1 2 v ; r = ( 3 ± 13) 4; saddle point, unstable 3. b., c. (0 , 0); u = u, v = 3 2 v ; r = 1 , 3 2 ; node, unstable ( 0 , 3 2 ) ; u = 1 2 u, v = 3 2 u 3 2 v ; r = 1 2 , 3 2 ; node, asymptotically stable (1 , 0); u = − u v, v = 1 2 v ; r = − 1 , 1 2 ; saddle point, unstable 4. b., c. (0 , 0); u = u, v = 5 2 v ; r = 1 , 5 2 ; node, unstable ( 0 , 5 3 ) ; u = 11 6 u, v = 5 12 u 5 2 v ; r = 11 6 , 5 2 ; saddle point, unstable (1 , 0); u = − u + 1 2 v, v = 11 4 v ; r = − 1 , 11 4 ; saddle point, unstable (2 , 2); u = − 2 u + v, v = 1 2 u 3 v ; r = ( 5 ± 3) 2; node, asymptotically stable 6. a. Critical points are x = 0, y = 0; x = 𝜖 1 𝜎 1 , y = 0; x = 0, y = 𝜖 2 𝜎 2 . x 0 , y 𝜖 2 𝜎 2 as t ; the redear survive. b. Same as part a except x 𝜖 1 𝜎 1 , y 0 as t ; the bluegill survive. 7. a. X = ( B 𝛾 1 R ) (1 𝛾 1 𝛾 2 ) , Y = ( R 𝛾 2 B ) (1 𝛾 1 𝛾 2 ) b.

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• Spring '16
• Anhaouy
• Districts of Vienna, Boyce, e2t, 3y, = min, + c2 sin x

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