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Answers to Odd-Numbered Problems CHAPTER 1 Exercises 1.1 1. (a) ordinary, frst order (c) partial, second order (e) ordinary, third order (g) ordinary, second order 3. Both y and z are solutions. 5. Both y and z are solutions. 7. Both u 1 and u 2 are solutions. 9. u 1 is a solution; u 2 is not a solution. 11. y =16 x 2 + C 1 x + C 2 13. y = Ce 3 x . 15. r =2 , - 2; y 1 ( x )= e 2 x and y 2 ( x e - 2 x are solutions. 17. r =3 ; y ( x e 3 x is a solution. 19. No real values oF r ; r =1 ± 2 i are complex values. 21. r , - 3; y 1 ( x x 3 and y 2 ( x x - 3 are solutions. Exercises 1.3 1. (b) y e 5 x . 3. (b) y = e e - 2 e x . 5. (b) y = - sin 3 x + 1 3 cos 3 x . 7. (b) y = - 17 4 x +9 x . (c) y ± is not defned at x = 0; there is no solution to y ± (0) = 2. 11. xy ± - 3 y +3=0. 13. y ± - 2 y = - 4 e - 2 x . 15. y ± 2 + xy ± - y =0 . 17. y ±± - 4 y ± +4 y . 1

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19. x 2 y ±± + xy ± - y =0 . 21. y ±± +9 y . 23. y ±±± . CHAPTER 2 Exercises 2.1 1. y = - 1 2 + Ce 2 x . 3. y =1+ - x 2 . 5. y = e - x + x . 7. y = x - 2 sin x + Cx - 2 . 9. y = 2 9 ( x +1) 5 / 2 + C ( x - 2 . 11. y = sin x cos x + C cos x = 1 2 sin 2 x + C cos x . 13. y = e x + C x . 15. y = x (ln x ) 2 + . 17. y - e x . 19. y = x - 1+2 e - x . 21. y = ln (1 + e x ) e x +( e - ln 2) e - x . 23. y = 5 - cos 2 x 2 sin x . 25. y = 2 - 3 x 3 . 27. y = ( 2 x - e x ) 2 . 29. y = 1 3 3 - 2 x 3 ln x . Exercises 2.2 1. y = ± x 2 4 + C ² 2 . 3. tan - 1 y = x 3 + C or y = tan ( x 3 + C ) . 5. cot y =ln ³ 1 - x 1+ x + C . 2
7. e - y = e x - xe x + C . 9. y = x + C 1 - Cx . 11. y 2 = C (ln x ) 2 - 1. 13. ln | y | = - ln | x |- 1 x - 1. 15. y = xe x 2 - 1 . 17. y +ln | y | = 1 3 x 3 - x - 5. 19. y 2 = C 1+ x 2 - 1. 21. y = ln | sec x + tan x | x + C x . 23. y =1+ Ce - x 2 . 25. y = C (3 x 2 +1) 1 / 3 - 3. 27. y = 1 +1+ln x . 29. y = x 2 + . 31. x ln x + x + y e y/x = . 33. csc( y/x ) - cot( y/x )= . Exercises 2.3.1 1. x 2 +3 y 2 = C . 3. x 2 2 + y 2 - 4 y = C . 5. x 2 2 + y 2 = C ; ellipses, center at the origin, major axis horizontal. 7. y = C ( x - a ) Exercises 2.3.2 1. (a) A ( t )=50 ± 9 10 ² t/ 2 50 e - 0 . 05268 t . (b) A (4) = 50 ( 9 10 ) 2 =40 . 5 grams. (c) T 13 . 16 hours. 3. (a) P ( t ) 0 . 25 e 0 . 0421 t (b) 1 . 6573 square centimeters (c) 16 . 464 hours 5. (a) P ( t ) 4 . 5 e 0 . 01438 t . (b) 48.19 years (c) 6 . 17 billion. 3

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Exercises 2.3.3 1. (a) 40 . 1 . (b) 1.62 minutes. 3. 8:52 pm. Exercises 2.3.4 1. (a) v = ± v 0 + g r ² e - rt - g r (b) lim t →∞ v = - g r . (c) y = y 0 + 1 r ± v 0 + g r ² ( 1 - e - ) - g r t 3. k 17 . 8 Exercises 2.3.5 1. (a) A ( t )=10 , 000 ( 1 - e - t/ 200 ) (b) t = 200 ln 5 322 minutes 3. (a) A ( t )= 9 2 ( 1 - e - t/ 150 ) (b) t = 150 ln 3 165 minutes 5. (a) A ( t 3 20 t (100 - t ) (b) max = A (50) = 375 Exercises 2.3.6 1. (a) 3259 people. (b) 6 . 89 days. 4
CHAPTER 3 Exercises 3.2 1. Yes 3. 5. 7. (a) r = - 1 ,r =4 . (b) Fundamental set: y 1 ( x )= x - 1 ,y 2 ( x x 4 ; general solution: y = C 1 x - 1 + C 2 x 4 . (c) y = 9 5 x - 1 + 1 5 x 4 . (d) The trivial solution: y 0. 9. y ±± - 2 y ± - 3 y =0. 11. y ±± =0 . 13. x 2 y ±± - 2 xy ± +2 y . 15. W [ y 1 2 ]( x e - ± x a p ( t ) dt ± = 0 for all x . 17. { y 1 ( x x, y 2 ( x x 2 } . 19. { y 1 ( x e x 2 2 ( x e - x 2 } . 21. αδ - βγ ± . 23. W [ y 1 + y 2 1 - y 2 ]= - 2 W [ y 1 2 ]. 25. Set u ( x y 2 ( x ) y 1 ( x ) . Then u ± ( x y 1 y ± 2 - y 2 y ± 1 y 2 1 = W [ y 1 2 ] y 2 1 0 . Therefore, u λ constant, which implies that y 2 = λy 1 . Exercises 3.3 1. y = C 1 e 2 x + C 2 e - 4 x . 3. y = C 1 e 5 x + C 2 xe 5 x . 5. y = e - 2 x [ C 1 cos 3 x + C 2 sin 3 x ]. 7. y = C 1 + C 2 e - 2 x . 9. y = C 1 e 2 3 x + C 2 e - 2 3 x . 11. y = e x [ C 1 cos x + C 2 sin x ].

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