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Y e 2 t for 0 t 1 y e t 1 for t 1 section 25 page 67

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y = e 2 t for 0 t 1; y = e ( t + 1) for t > 1 Section 2.5, page 67 1. y = − a b is asymptotically stable, y = 0 is unstable 2. y = 1 is asymptotically stable, y = 0 and y = 2 are unstable 3. y = 0 is unstable 4. y = 0 is asymptotically stable 5. c. y = ( y 0 + (1 y 0 ) kt ) (1 + (1 y 0 ) kt ) 6. y = − 1 is asymptotically stable, y = 0 is semistable, y = 1 is unstable 7. y = − 1 and y = 1 are asymptotically stable, y = 0 is unstable 8. y = 2 is asymptotically stable, y = 0 is semistable, y = − 2 is unstable 9. y = 0 and y = 1 are semistable 16. a. y = 0 is unstable, y = K is asymptotically stable b. Concave up for 0 < y K e , concave down for K e y < K 17. a. y = K exp (( ln ( y 0 K )) e rt ) b. y (2) 0 . 7153 K 57 . 6 × 10 6 kg c. 𝜏 2 . 215 yr 18. b. V = ( k ( 𝛼 𝜋 )) 3 2 𝜋 h (3 a ); yes c. k < 𝛼 𝜋 a 2 19. c. Y = Ey 2 = KE (1 ( E r )) d. Y m = Kr 4 for E = r 2 20. a. y 1 , 2 = K (1 1 (4 h rK )) 2 21. a. y = 0 is unstable, y = 1 is asymptotically stable b. y = y 0 ( y 0 + (1 y 0 ) e 𝛼 t ) 22. a. y = y 0 e 𝛽 t b. x = x 0 exp ( 𝛼 y 0 (1 e 𝛽 t ) 𝛽 ) c. x 0 exp ( 𝛼 y 0 𝛽 )
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Boyce 9131 BMAnswersToProblems 2 March 11, 2017 15:55 576 576 Answers to Problems 23. b. z = 1 ( 𝜈 + (1 𝜈 ) e 𝛽 t ) c. 0.0927 24. a, b. a = 0: y = 0 is semistable. a > 0: y = a is asymptotically stable and y = − a is unstable. 25. a. a 0: y = 0 is asymptotically stable. a > 0: y = 0 is unstable; y = a and y = − a are asymptotically stable. 26. a. a < 0: y = 0 is asymptotically stable and y = a is unstable. a = 0: y = 0 is semistable. a > 0: y = 0 is unstable and y = a is asymptotically stable. 27. a. lim t x ( t ) = min ( p, q ); x ( t ) = pq ( e 𝛼 ( q p ) t 1) qe 𝛼 ( q p ) t p b. lim t x ( t ) = p ; x ( t ) = p 2 𝛼 t p 𝛼 t + 1 Section 2.6, page 75 1. x 2 + 3 x + y 2 2 y = c 2. Not exact 3. x 3 x 2 y + 2 x + 2 y 3 + 3 y = c 4. ax 2 + 2 bxy + cy 2 = k 5. Not exact 6. e xy cos 2 x + x 2 3 y = c 7. y ln x + 3 x 2 2 y = c 8. x 2 + y 2 = c 9. y = ( x + 28 3 x 2 ) 2 , | x | < 28 3 10. y = ( x (24 x 3 + x 2 8 x 16) 1 2 ) 4 , x > 0 . 9846 11. b = 3; x 2 y 2 + 2 x 3 y = c 12. b = 1; e 2 xy + x 2 = c 15. x 2 + 2 ln | y | y 2 = c ; also y = 0 16. x 2 e x sin y = c 18. 𝜇 ( x ) = e 3 x ; ( 3 x 2 y + y 3 ) e 3 x = c 19. 𝜇 ( x ) = e x ; y = ce x + 1 + e 2 x 20. 𝜇 ( y ) = y ; xy + y cos y − sin y = c 21. 𝜇 ( y ) = e 2 y y ; xe 2 y − ln | y | = c ; also y = 0 Section 2.7, page 82 1. a. 1.2, 1.39, 1.571, 1.7439 b. 1.1975, 1.38549, 1.56491, 1.73658 c. 1.19631, 1.38335, 1.56200, 1.73308 d. 1.19516, 1.38127, 1.55918, 1.72968 2. a. 1.1, 1.22, 1.364, 1.5368 b. 1.105, 1.23205, 1.38578, 1.57179 c. 1.10775, 1.23873, 1.39793, 1.59144 d. 1.1107, 1.24591, 1.41106, 1.61277 3. a. 1.25, 1.54, 1.878, 2.2736 b. 1.26, 1.5641, 1.92156, 2.34359 c. 1.26551, 1.57746, 1.94586, 2.38287 d. 1.2714, 1.59182, 1.97212, 2.42554 4. a. 0.3, 0.538501, 0.724821, 0.866458 b. 0.284813, 0.513339, 0.693451, 0.831571 c. 0.277920, 0.501813, 0.678949, 0.815302 d. 0.271428, 0.490897, 0.665142, 0.799729 5. Converge for y 0; undefined for y < 0 6. Converge for y 0; diverge for y < 0 7. Converge for | y (0) | < 2 . 37 (approximately); diverge otherwise 8. Diverge 9. a. 2.30800, 2.49006, 2.60023, 2.66773, 2.70939, 2.73521 b. 2.30167, 2.48263, 2.59352, 2.66227, 2.70519, 2.73209 c. 2.29864, 2.47903, 2.59024, 2.65958, 2.70310, 2.73053 d. 2.29686, 2.47691, 2.58830, 2.65798, 2.70185, 2.72959 10. a. 1.70308, 3.06605, 2.44030, 1.77204, 1.37348, 1.11925 b. 1.79548, 3.06051, 2.43292, 1.77807, 1.37795, 1.12191 c. 1.84579, 3.05769, 2.42905, 1.78074, 1.38017, 1.12328 d. 1.87734, 3.05607, 2.42672, 1.78224, 1.38150, 1.12411 11. a. 0.166134, 0.410872, 0.804660, 4.15867 b. 0.174652, 0.434238, 0.889140, 3.09810 12. A reasonable estimate for y at t = 0 . 8 is between 5.5 and 6. No reliable estimate is possible at t = 1 from the specified data.
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  • Spring '16
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  • Districts of Vienna, Boyce, e2t, 3y, = min, + c2 sin x

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