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Fourth order linear 4 second order nonlinear 11 r 2

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Fourth order, linear 4. Second order, nonlinear 11. r = − 2 12. r = 2 , 3 13. r = 0 , 1 , 2 14. r = − 1 , 2 15. r = 1 , 4 16. Second order, linear 573 f
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Boyce 9131 BMAnswersToProblems 2 March 11, 2017 15:55 574 574 Answers to Problems 17. Fourth order, linear 18. Second order, nonlinear Chapter 2 Section 2.1, page 31 1. c. y = ce 3 t + t 3 1 9 + e 2 t ; y is asymptotic to t 3 1 9 as t 2. c. y = ce 2 t + t 3 e 2 t 3; y as t 3. c. y = ce t + 1 + t 2 e t 2; y 1 as t 4. c. y = c t + 3 cos (2 t ) (4 t ) + 3 sin (2 t ) 2; y is asymptotic to 3 sin (2 t ) 2 as t 5. c. y = ce 2 t 3 e t ; y or −∞ as t 6. c. y = − te t + ct ; y , 0 , or −∞ as t 7. c. y = ce t + sin (2 t ) 2 cos (2 t ); y is asymptotic to sin (2 t ) 2 cos (2 t ) as t 8. c. y = ce t 2 + 3 t 2 12 t + 24; y is asymptotic to 3 t 2 12 t + 24 as t 9. y = 3 e t + 2( t 1) e 2 t 10. y = ( t 2 1) e 2 t 2 11. y = ( sin t ) t 2 12. y = ( t 1 + 2 e t ) t, t 0 13. b. y = − 4 5 cos t + 8 5 sin t + ( a + 4 5 ) e t 2 ; a 0 = − 4 5 c. y oscillates for a = a 0 14. b. y = ( 2 + a (3 𝜋 + 4) e 2 t 3 2 e 𝜋 t 2 ) (3 𝜋 + 4); a 0 = − 2 (3 𝜋 + 4) c. y 0 for a = a 0 15. b. y = te t + ( ea 1) e t t ; a 0 = 1 e c. y 0 as t 0 for a = a 0 16. b. y = ( e t e + a sin 1 ) ∕ sin t ; a 0 = ( e 1) ∕ sin 1 c. y 1 for a = a 0 17. ( t, y ) = (1 . 364312 , 0 . 820082) 18. y 0 = − 1 . 642876 19. a. y = 12 + 8 65 cos 2 t + 64 65 sin 2 t 788 65 e t 4 ; y oscillates about 12 as t b. t = 10 . 065778 20. y 0 = − 5 2 21. y 0 = − 16 3; y −∞ as t for y 0 = − 16 3 29. See Problem 2. 30. See Problem 4. Section 2.2, page 38 1. 3 y 2 2 x 3 = c ; y 0 2. y 1 + cos x = c if y 0; also y = 0; everywhere 3. 2 tan (2 y ) 2 x − sin (2 x ) = c if cos (2 y ) 0; also y = ± (2 n + 1) 𝜋 4 for any integer n ; everywhere 4. y = sin ( ln | x | + c ) if x 0 and | y | < 1; also y = ± 1 5. y 2 x 2 + 2( e y e x ) = c ; y + e y 0 6. 3 y + y 3 x 3 = c ; everywhere 7. y = kx 8. y = ± x 2 + c 9. a. y = 1 ( x 2 x 6) c. 2 < x < 3 10. a. y = − 2 x 2 x 2 + 4 c. 1 < x < 2 11. a. y = (2(1 x ) e x 1) 1 2 c. 1 . 68 < x < 0 . 77 approximately 12. a. r = 2 (1 2 ln 𝜃 ) c. 0 < 𝜃 < e 13. a. y = ( 3 2 1 + x 2 ) 1 2 c. | x | < 1 2 5 14. a. y = − 1 2 + 1 2 4 x 2 15 c. x > 1 2 15 15. a. y = 5 2 x 3 e x + 13 4 c. 1 . 4445 < x < 4 . 6297 approximately 16. a. y = ( 𝜋 arcsin ( 3 cos 2 x )) 3 c. | x 𝜋 2 | < 0 . 6155 17. y 3 3 y 2 x x 3 + 2 = 0 , | x | < 1 18. y 3 4 y x 3 = − 1 , | x 3 1 | < 16 3 3 or 1 . 28 < x < 1 . 60 19. y = − 1 ( x 2 2 + 2 x 1); x = − 2 20. y = − 3 2 + 2 x e x + 13 4; x = ln (2) 21. a. y 4 if y 0 > 0; y = 0 if y 0 = 0; y −∞ if y 0 < 0 b. T = 3 . 29527 22. a. y 4 as t b. T = 2 . 84367 c. 3 . 6622 < y 0 < 4 . 4042 23. x = c a y + ad bc a 2 ln | ay + b | + k ; a 0 , ay + b 0 25. c. | y + 2 x | 3 | y 2 x | = c 26. b. arctan ( y x ) − ln | x | = c 27. b. x 2 + y 2 cx 3 = 0 28. b.
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