Answers to Selected Exercises
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- ψυχή Στεφανόπουλος
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Answers to Selected Eercises Chapter. second, fifth, fifth, forty-second a i. yes, it is a ii. no, it is not a iii. no b i. no b ii. yes b iii. no c i. yes c ii. no c iii. no d i. no d ii. no d iii. yes e i. no e ii. yes e iii. yes f i. no f ii. no f iii. yes g i. no g ii. yes g iii. yes h i. yes h ii. yes h iii. yes a i. no a ii. yes a iii. no b i. no b ii. no b iii. yes c i. yes c ii. no c iii. no d i. no d ii. no d iii. yes 5b i. 9 5b ii. 5 5c i. y = + 9 5c ii. y = + 5 6b i. y = e 6b ii. y = e 7b i. y = cos) + sin) 7b ii. y = sin) 8a i. 7 8a ii. No, because then 7 = y ) =. d 8a iii. y = d d + y +, y ) = 9 8a iv. Of course. 8b i. y 0) = and y 0) = 8b ii. No. 9a. 00/7 sec. 9b. 0 meters/second 9c i. y 0) = 0 is replaced with y 0) = 9c ii..9t + t c iii. dv 90 meters/second 0a. /dt = 9.8 κv ) 0b i. 000κ 0b ii. v hit + 9.8t hit )/000 b. + 5 ln c. Because y) is not continuous at = 0. Chapter a. Yes, it is directly integrable. b. No, it is not directly integrable. c. No d. Yes e. No f. Yes g. Yes h. No i. No a. + c b. 5e + c c. ln + c d. + + c e. sin ) + c f. sin) + cos) + c g. ln 9 + c h. 6 ln + + c i c j. sin) + c + c k c + c l. + c + c + c + c a. + 5e for < < b. + 6) / + for < < c. ln for < < d. ln + for 0 < < e. ln cos) + for π < < π f. arctan) + for < < g. ln ) for 0 < < 5a. cos + + y0) 5b i. 5b ii. 5 5b iii. 7 6a. + ) / + y) 6 6b i. 5 6b ii
2 66 Answers to Selected Eercises 7 6b iii. 7a. e 7b c. arctan π { } π 7d. 6 erf) + 7e. Si) + 7f. Si ) 0 if < 0 ) 9a. if 0 { } if < 0 if < 9b. 9c. if + if if < 9d. + c if s + + c if < s Chapter a. = 6 y, nonautonomous, const. solns.: y = d sin + y) b. =, nonautonomous, const. solns.: none d y c. d = y + 8, autonomous, const. solns.: y = d. d = y, nonautonomous, const. solns.: y = and y = e. d = + y, nonautonomous, const. solns.: none f. d = 5y y, autonomous, const. solns.: y = 0, y = 5 and y = 5 g. d = y +, nonautonomous, const. solns.: y = h. d =, nonautonomous, const. solns.: none y i. d = y y, autonomous, const. solns.: y = + and y = j. d = y + 8)y 8, nonautonomous, const. solns.: y = 8 5b. Because F / y with F, y) = y is not continuous at the point, 0). 6a. + 6b c a b Chapter a. d = sin))y b. Not separable c. Not separable d. Separable e. d = y) f. = y) g. Not separable d h. d = )y ) i. Not separable j. d = e y e y a. y = ± + c b. y = tan + c) c. y = ± A 9 d. y = tanc + arctan)) e. y = arcsinc cos)) f. y = ln e + c 5a. y = + 8 5b. y = e 5c. y = 5e / 5d. y = + ) 9 6a. y = 6b. y = / 6c. y = and y = 6d. y = 0,±π,±π,±π,... 6e. No constant solution 6f. y = 0 and y = 00 7a. y = + A ep ) 7b. y = c cos) ) and y = 0
3 Answers to Selected Eercises 67 ) 7c. y = + A ep 7d. y = arcsin Ae ) 7e. y = A 7f. y y = + + c 7g. y = tanarctan) + c) 7h. implicit: y + y = + c, eplicit: y = c ± c ) ; also y = 0 7i. y = ln + c 7j. y = ln Ae 7k. y = ± c ) / and y = 0 7l. y + y / = + / + c 7m. y = ± + Ae 7n. y = tan + c ) 7o. y = 00Ae00 + Ae 00 8a. y = 5 + e 8b. y = 8 cos) 8c. y = ) 8d. y = 0 8e. y ln y = + 8f. y = + 9a. y = 0 and y = β β / γ 9b. y = γ Ae β and y = 0 0a., ) ) 0b., ) 0c., ) 0d. e, ) 0e., Chapter 5 a. d + y = sin) b. Not linear c. Not Linear d. Not Linear e. d + )y = f. d y = 8 g. d + 0 y = e h. d sin) y = 0 i. Not linear j. d 87 y = cos ) a. + ce b. e + ce c. ce d. ce e. + c f. c cos ] g. c 5 h. + c] cos) i. + ce 5] j. / + ce a. 7e b. c. e e 5] d. + 6 e. sin) ] f ) ] / a. e + e s sins) ds b. 0 + sins) s ] ds c a. µ ) = µ)p) and y ) = f ) p) µ) Chapter 6 0 ] e s ds µ 0 )y ] µs) f s) ds a i. + y + = tany + C) a ii. y = ± Ae a iii. y = + arcsin8 + c) b. y = a i. y = ln + c) and y = 0 a ii. y = ± c + ln a iii. y = arcsina ) b. y = a i. y = ± + ce 6) / and y = 0 a ii. y = C ) and y = 0 a iii. y = sin) + c ) and y = 0 b. y = sin) a. u = y / ; y = ln + c) / b. u = + y + ; y = + c ) and y = + ) c. u = y / ; y = + c / ) and y = 0 d. u = y ; y = arccos + c) e. u = y ; y + ln y + = c and y = + f. u = y / ; y ln y cy = and y = 0
4 68 Answers to Selected Eercises g. u = y / ; y = ± ln + c h. u = y ; y = ± c ) / and y = 0 i. u = + y ; y = + c) + and y = j. u = + y ; + y ln + ) + y = + c ) k. u = y / ; y = ln + c and y = l. u = y ; y = ± 8e + ce ) / m. u = y + ; y = + + Ae ) Ae + ) and y = + n. u = y + ; y = c + c and y = o. u = siny) ; y = arcsin c + )e ) p. u = y ; y = tanc + ln ) 5. du /d + n)p)u = n) f ) Chapter 7 a. y + d = 0, y = c b. 6 y + y ] d = 0, y = ± c + 6 c. ] y y + y ] d = 0, y y = c d. Arctany) + c ) + y d = 0, y = tan b. y = ± + c a. φ, y) = y + y, y = ± + C b. φ, y) = y +, y = c ) / c. φ, y) = + y, y = c + ) / d. φ, y) = y + + y c, y = ± e. φ, y) = y + 5 y5, y 5 y5 = c f. φ, y) = ln y, y = e c g. φ, y) = + e y, y = ln c h. φ, y) = e y + y, e y + y = c 5a. µ =, y y = c 5b. µ = y, y + y = c 5c. µ = y, y y = c 5d. µ = cosy), cosy) + siny) = c 5e. µ =, y = ± + C 5f. µ = e, y = Ce 5g. µ =, y = ± c ) / 5h. µ = y, y 5/ + y / = c 5i. µ = y /, y 7 / + y / = c Chapter 8 b. y0) should be within / 6 of.. a., b. y8) /
5 Answers to Selected Eercises 69. a., bi. y) /, bii. y), biii. y) 6 / 5. ai. & bi., aii. The ma. is approimately 6 / and is at., aiii. y0) /, bii. y0) / 6. ai. & bi., aii. The ma. is approimately 5 and is at., aiii. y0) /, aii. The ma. is approimately and is at 6 /., bii. y0) / 7. ai. & bi., aii. y) /, bii. y0) 9a. y = appears to be an asymptotically stable constant solution. 9b. y = appears to be an unstable constant solution. 9c. y = appears to be a stable but maybe not asymptotically stable) constant solution. 9d. y = appears to be an unstable constant solution. 9e. y = appears to be an unstable constant solution, and y = appears to be an asymptotically stable constant solution. 9f. y = appears to be a stable but not asymptotically stable) constant solution.
6 60 Answers to Selected Eercises Chapter 9 a. k k y k 0 / / 5 / 5 / b. k k y k / 0 / 5 / 0 0 c. k k y k 0 0 / / 9 / 985 /8 d. k k y k 0 8 / / / 6 /5 5 7 / /7 a. k k y k b. k k y k c. k k y k d. k k y k a..088 b c d a. y) with error =.0568 ± b. y) with error =.5976 ± c. y) with error =.0097 ± d. y) with error =.56 ± e. y) 0.9 with error =.05 ± f. y) 0.57 with error =.00 ± g. y) with error =.000 ± a. y) =, y) = c i. y) with error = 99.8 ± c ii. y). with error =.69 ± c iii. y) with error = 0.0 ± c iv. y) with error = 0.00 ± a i. y k = ) k 7a ii. It does not. 7a iii. y) 0 as ; y k y k ) 7b i. y k = 7b ii. yes 8d. The solution becomes infinite. 8e. They are nonsense. The solution to the initial-value problem is not valid for > 7 /. Chapter 0 ) k 5 a. 6,58,05 b. for each, the answer is 0.8 ln 0.55) month c. for each, the answer is 0.8 ln 0.8) months d.. months a. It) = e βt with β = ln50) b.,500 c..78 days 5a. 0.0/year 0 5b i grams 5b ii.. grams 5b iii..98 grams 6a /year 6b i % 6b ii % 6b iii % 6b iv. 5.6 % 6b v. 9.8 %
7 Answers to Selected Eercises 6 6b vi. 0. % 6c. 9,95 years ago 6d., years ago 6e. 570 ) ln ln A A 0 7b i. 6.8 rabbits 7b ii., 95,.8 rabbits 7b iii. 9, 999, 995. rabbits e 7c. κ = R 0 Rt) βt R 0 Rt)e βt 7d i.,8.5 rabbits 7d ii.,7.6 rabbits d R 8a. = 5 R 500 8b. y = 00 is the equilibrium solution. If we start with more, dt the population increases. If we start with less, the population decreases rapidly!) 8c. Rt) = 00 + R 0 00)e 5t/ d R 9a. = R 9b. There is no equilibrium dt solution; if we start with a positive number of rabbits, the population constantly increases. 9c. Rt) = R 0 e t/ d R 0a. = β R γ R h 0 dt d R 0b. = β ) R γ R a. = 50 y initial condition: y0) = 00 ) dt dt 0 b. 000 warriors c. yt) = e t/0 d. 0 ln 8 weeks about.5 weeks) a. = 9 dt 000 y c. y = e t/000 d i.. gallons d ii..6 gallons d iii. 7.7 gallons e. When t = 000 ln 66. a. = 9 dt y 000 c. y = e t/000 d i. 9. gallons d ii. 07. gallons d iii gallons e. When t = 000 ln a i. = dt 00 y a iii. yt) = e t/00 b i..8 b ii. 8.6 b iii..7 c. oz. salt/gal. water d. When t = 00 ln 9. 5a i. = dt 00 y 5a iii. yt) = 00 00e t/00 5b i..6 5b ii b iii c. / oz. salt/gal. water 5d. Never 6a. 500 t 6b i. = dt 500 t y 6b ii. y = 500 t) 0 t ] 6c i c ii c iii d i. t = 99 6d ii ounces 7. Arrest the butler. The time of death was only about about :0, over an hour after the butler reported finding the bo. Besides, his fingerprints were on the bottle. Chapter a. y = + c + c b. y = A + c c. y = Ae + c d. y = e + c e + c e. y = Ae + c f. y = C arctan) + C a. y = c + c + c b. Equation contains y. c. y = c ± + c ) / d. y = ln + A + C, y = c and y = + c A e. y = 6 + c + c f. Equation contains y. g. y = + Ae + C h. Equation contains y. i. y = e + Ae + B j. y = B ln A + e a. y = Ae + B + C b. y = A ln + B + C c. y = + A) + B + C d. y = Ae + B + C + D a. y = Be a b. y = A + c) c. y = arccosc A) d. y = Ae + c e. y = ± Ae + C f. y + y = A + C 5a. not autonomous 5b. y = ± A + c 5c. y = c ± + c ) / 5d. not autonomous 5e. not autonomous 5f. y = Be A + A and y = c 5g. y = + Ae + C 5h. y = + Be A 5i. not autonomous 5j. y = ln A + Be
8 6 Answers to Selected Eercises 6a i. y = a ii. y = + 8 6a iii. y = 5e + 6a iv. y = e + e 6a v. y = e a vi. y = ln + + 6b i. y = + 6b ii. y = + ) / 7a i. y = 5e 7a ii. y = 8 + ) 7a iii. y = ) 7b i. y = e + 5 7b ii. y = ln e 8a i. y = arctan) + 8a ii. y = 8a iii. y = 8a iv. y = ln b. y = A arctana) + c, y = c and y = a ln a + c 9a i. y = tan) + a 9a ii. y = 9a iii. y = 9a iv. y = e + e 9b. y = A tana + B), y = c Chapter and y = c + Bec Be c a. second-order, linear, nonhomogeneous b. second-order, linear, homogeneous c. second-order, linear, homogeneous d. second-order, nonlinear e. first-order, linear, nonhomogeneous f. third-order, linear, homogeneous g. second-order, nonlinear h. second-order, linear, nonhomogeneous i. fourth-order, linear, nonhomogeneous j. third-order, nonlinear k. third-order, linear, homogeneous l. fifty-fifth-order, linear, nonhomogeneous a. L = d d + 5d d + 6 b i. 5 sin) + 5 cos) b ii. e b iii. 0 b iv c. e a. L = d d 5d d + 9 b i. 8 sin) 5 cos) b ii. 5 cos) b iii. e b iv. sin) cos)] e 5a. L = d 5b i. 6 sin) + 5 cos) sin) d + 5 d d + 6 5b ii ] e 5b iii. 7 6a. L = d d sin) d + cos) 6b i. cos) 6b ii. + sin) d 6b iii. cos) sin) 7a. y) = cos) + sin) 7b. y) = e e 7c. y) = e + 5e 7d. y) = e + e 7e. y) = 5 / + / 7f. y) = 5 ln 7g. y) = cos ) sin ) π 7h. y) = + 8a. y) = cos) + sin) 8b. y) = + sin ) + 8 sin) cos) 8c. y) = sin) + sinh) a. L L = d d ], L L = d d + ] b. L L = d d + + ] d d + + 5], L L = d d + + ] d d + + 5] c. L L = d d + + ] d d + 6, L L = d d + + ] d d + 8 d. L L = d d L L = d d + 6 d d + 6, L L = d d + d d e. L L = d d, f. L L = sin) d d, L L = sin) d d + cos)d d sin) a. d d + 5d d + 6
9 Answers to Selected Eercises 6 b. d d + 6 d d + 6 c. d d + 5d d + d d. d + + ] d d + ] d e. d + + ] d d + 8 d f. d + 0 d d ] d g. d + + ] d d + d d d h. d + + ] d d + + ] d + + ]. y) = ce d. y) = ce 5. y) = c Chapter a. Ae + Be b. Ae 5 + Be 5 c. A + B d. A ln + B e. A e + Be f. Ae + Be g. A cos) + B sin) h. A e + B i. A sin) + B sin) j. A cos) + B sin) k. A cosln ) + B sinln ) l. A / cos) + B / sin) a. Ae + Be e b. Ae + Be e c. A + B d. A 5 + B + 5 ln e. y = e + A + Be ] f. y = + + B + Ae a. y = Ae + Be + C e b. y = e cos) + Ae + Be + C e c. y = Ae + Be + C e + D e d. y = A ln + B + C Chapter a. gen. soln.: y) = c e + c e, i.v. soln.: y) = e + e b. gen. soln.: y) = c e + c e, i.v. soln.: y) = e e c. gen. soln.: y) = c e + c e 9, i.v. soln.: y) = e e 9 d. gen. soln.: y) = c + c e 5, i.v. soln.: y) = a. c + c e + c e b. c e + c e + c e + c e Chapter 5 c. Yes, {y, y } is linearly dependent on any subinterval of either 0, ) or, 0). Chapter 6 a. Ae + Be 5 5 a i. e e5 a ii. e + e5 a iii. e + e 5 b. Ae + Be 6 b i. 5 e + 5 e 6 b ii. 7e + e 6 c. Ae 5 + Be 5 c i. e5 + e 5 c ii. 0 e5 0 e 5 c iii. e 5 + e 5 d. A + Be d i. d ii. e d iii. 6 + e a. Ae 5 + Be 5 a i. e 5 5e 5 a ii. e 5 a iii. e 5 + e 5 b. Ae + Be b i. e + e b ii. e b iii. 8e + 0e c. Ae / + Be / c i. e / e/ c ii. e / c iii. 6e / e / a. A cos5) + B sin5) a i. cos5) a ii. 5 sin5) a iii. cos5) sin5) b. Ae cos) + Be sin) b i. e cos) + e sin) b ii. e sin)
10 6 Answers to Selected Eercises b iii. 5e cos) + 7e sin) a. y) = e cos) b. y) = e cos) 5a. y) = c e + c e 5b. y) = c cos) + c sin) 5c. y) = c e + c e 5d. y) = c e + ) + c e ) 5e. y) = c e / + c e / 5f. y) = c e cos) + c e sin) 5g. y) = c e cos6) + c e sin6) 5h. y) = c e + c e / 5i. y) = c e 5 + c) e 5 ) 5j. y) = c e / + c e / 5k. y) = c cos + c sin ) ) 5l. y) = c + c e /9 5m. y) = c e cos + c e sin 5n. y) = c e cos) + c e sin) 5o. y) = c e + c e 5p. y) = c e + c e 5 5q. y) = c + c e ) ) 5r. y) = c e + c e 5s. y) = c cos + c sin 5t. y) = c e / cos) + c e / sin) Chapter 7. kg/sec a. kg/sec ) b. ω 0 =, ν 0 = π, p 0 = π c i. A =, φ = 0 c ii. A =, φ = π c iii. A =, φ = π c iv. A =, φ = π a. 88 kg/sec ) b. ω 0 =, ν 0 = 6 π, p 0 = π 6 c i. A = c ii. A = 7 π c iii. A = 5a. 8π 5b. π 5c b. α =, ω =, p = π, ν = 7c i. A = 7c ii. A = 7c iii. A = 5 π 9. ν decreases from the natural frequency of the undamped system, ν 0, down to zero. p increases from the natural period of the undamped system, p 0, up to. 0b. yt) = + t]e t 0c. yt) = te t b. yt) = 8 e t e t c. yt) = e t e t Chapter 8 a. c + c + c + c e b. c + c + c cos) + c sin) c. c e + c e + c e 5 + c e 5 d. c e + c e + c cos) + c sin) e. c + c ]e + c + c ]e f. c + c + c ] cos) + c + c 5 ] sin) a. c e + c cos) + c sin) b. c e + c e + c e c. c e + c e cos) + c e sin) d. c + c ]e + c cos) + c sin) a. 6 cos) + sin) b. ] e c. cos) sin) + cos5) 5 sin5) ) ) a. c e + c e cos + c e sin b. c cos) + c sin) + c cos) + c sin) c. c + c + c ] e + c + c 5 + c 6 ] e d. c + c ]e + c + c ]e / cos Chapter 9 ) + c 5 + c 6 ]e / sin a. y = c + c b. y = c + c c. y = c + c d. y = c + c e. y = c + c ln f. y = c + c ln )
11 Answers to Selected Eercises 65 g. y = c + c ln h. y = c cos ln ) + c sin ln ) i. y = c cos5 ln ) + c sin5 ln ) j. y = c cosln ) + c sinln ) k. y = c / + c l. y = c cos ln ) + c cos ln ) m. y = c + c ln n. y = c 5 + c 5 a. y = 5 b. y = / 6 / c. y = 6 6 ln d. y = 9 cos ln ) 5 sin ln ) a. y = c + c + c b. y = c + c cosln ) + c sinln ) c. y = c + c + c ln d. y = c + c + c + c e. y = c + c ln + c ln ) + c ln ) 5c. Replace each k y k) in the differential equation with rr )r ) r k ]) Chapter 0 a. 0e b. 9 ) e c. 0e a b. c. a. No, because y + y = 0 when y) = sin). b. cos) b. c cos) + c sin) c. e + c cos) + c sin) d i. e + cos) d ii. e 5 cos) sin) 5b. c e + c e 5c. + c e + c e 5d i. + e e 5d ii. + e + e 6b. + c e + c e 6c. + 7e + 5e 7b. e + c e 5 + c e 7c. e + e 5 + e 8b. e 5 + c e 5 + c e 8c. e 5 + e 5 + 9e 9b. 5 sin) cos) + c e + c e 9c. 5 sin) cos) + e + 5e 0b c + c 0c b. + c + c + c cos) + c sin) c cos) sin) a. y p ) = 7 e5, y) = 7 e5 + c 5 + c b. y p ) = e 5 + e, y) = e 5 + e + c 5 + c c. y p ) = 5e 5 e, y) = 5e 5 e + c 5 + c a i. g) = a ii. g) = 8 a iii. g) = 5 b i. y p ) = + + 5, y) = c + c 5 b ii. y p ) =, y) = + c + c 5 a i. g) = cos) + sin) a ii. g) = cos) sin) b i. y p ) = 5 cos) 5 sin), y) = 5 cos) 5 sin) + c e + c e b ii. y p ) = 5 cos) 5 sin), y) = 5 cos) 5 sin) + c e + c e Chapter a i. y p ) = e, y) = e + c cos) + c sin) a ii. y p ) = e 6, y) = e 6 + c e + c e a iii. y p ) = 6e, y) = 6e + c e + c e 5 a iv. y p ) = 7 e/, y) = 7 e/ + c + c e b. y) = e + e + e5 a i. y p ) = cos) + sin), y) = cos) + sin) + c cos) + c sin) a ii. y p ) = 9 cos6) sin6), y) = 9 cos6) sin6) + c e + c e
12 66 Answers to Selected Eercises a iii. y p ) = 9 cos ) + 7 sin ), y) = 9 cos ) + 7 sin + c + c e ) a iv. y p ) = sin) cos), y) = 6 sin) 6 cos) + c e + c e 5 b. y) = cos) sin) + 6e + e5 a i. y p ) = 0, y) = 0 + c e + c e 5 a ii. y p ) = , y) = c e +c e 5 a iii. y p ) = + +, y) = c e + c e a iv. y p ) = 9 7, y) = c cos) + c sin) b. y) = sin) a i. y p ) = 5 cos)+ sin), y) = 5 cos)+ sin)+c cos)+c sin) a ii. y p ) = e cos), y) = e cos) + c e + c e a iii. y p ) = e e + e, y) = e e + e +c cos)+c sin) a iv. y p ) = sin) cos) + cos)]e, y) = sin) cos) + cos)]e + c + c a v. y p ) = cos) + sin), y) = cos) + sin) + c e + c e a vi. y p ) = e5, y) = e5 + c e + c e b. y) = e e + 5 cos) 5 sin) 5a. y p ) = 7 e, y) = 7 e + c e + c e 5 5b. y p ) = 5, y) = 5 + c + c e 5c. y p ) = 6 +, y) = c + c e 5d. y p ) = cos), y) = cos) + c cos) + c sin) 5e. y p ) = 5 e, y) = 5 e + c e + c e 5f. y p ) = e e, y) = e e + c + c e 6a. y) = 6 ) e + c e + c e 5 6b. y) = 9 ) e 6 + c e + c e 5 6c. y) = 6 50 e 5 + c e 5 + c e 5 6d. y) = e 5 + c e 5 + c e 5 6e. y) = 9 5 cos) 5 sin) + c e cos) + c e sin) 6f. y) = e + c e cos) + c e sin) 6g. y) = e cos) + c e cos) + c e sin) 6h. y) = 9 e cos) + e sin) + c e cos) + c e sin) 6i. y) = + )e cos) + + 5)e sin) + c e cos) + c e sin) 6j. y) = 0 + c e cos) + c e sin) 6k. y) = 0 e + c e cos) + c e sin) 6l. y) = c e cos) + c e sin) 6m. y) = 0 e sin) 0 e cos) + c cos) + c sin) 6n. y) = cos) + sin) + c cos) + c sin) 6o. y) = cos) + sin) + c cos) + c sin) 7a. y p ) = ] A 0 + A + A + A e sin) + ] B 0 + B + B + B e cos)
13 Answers to Selected Eercises 67 7b. y p ) = A 0 + A + A + A ] e sin) + B 0 + B + B + B ] e cos) 7c. y p ) = A + B + C ] e 7 7d. y p ) = A + B + C 7e. y p ) = Ae 8 7f. y p ) = Ae 7g. y p ) = A + B + C ] e 7h. y p ) = A + B + C ] cos) + D + E + F ] sin) 7i. y p ) = A + B + C ] e cos) + D + E + F ] e sin) 7j. y p ) = Ae sin) + Be sin) 7k. y p ) = Ae sin) + Be cos) 7l. y p ) = A + B + C + D ] sin) + E + F + G + H ] cos) 8a. y p ) = e 8b. y p ) = e 8c. y p ) = 8 sin) cos) 8d. y p ) = 8e. y p ) = 8f. y p ) = cos) sin) 8g. y p ) = e 9a. y p ) = A + B + C ] e 9b. y p ) = A 5 + B + C ] cos) + D 5 + E + F ] sin) 9c. y p ) = A + B + C ] e cos) + D + E + F ] e sin) 9d. y p ) = A + B] cos) + C + D] sin) 9e. y p ) = A + B ] cos) + C + D ] sin) 9f. y p ) = A + B] e cos) + C + D] e sin) 9g. y p ) = A 5 + B + C + D + E + F ] e 0a. y p ) = e cos6) sin6) 0b. y p ) = 5 cos) + sin) cos) 0c. y p ) = 6 cos) + sin) 0d. y p ) = 5e e equiv., y p ) = 6 sinh) + cosh) ) Chapter a..9 kg/sec b. ω 0 = 7 0 /sec, ν 0 = 7 0 hertz a i. 50 kg/sec pi 98 a ii. 98 /sec) a iii..58 times per second b i..99 meter π 5 b ii. 5 /sec b iii. 5.6 times per second c. 7.5 kg π d. 9 kg meter/sec, a i. 6π t sinπt) a ii. t = 8π 5. seconds) b i. 7π cos6πt) b ii. No. The amplitude of the oscillations is less than.00 meter. c. 6 δ µ 6 + δ where δ = π F 0 5a. y η t) = m ω 0 ) η ] cosηt) cosω 0 t) ] 5b. y η t) = F 0 t sinω 0 t) mω 0 7b. No Chapter a. y = + c + c b. y = sin) ln csc) cot) + c cos) + c sin) c. y = cos) + sin) ln sin) + c cos) + c sin) d. y = e + c e + c e 5 e. y = + ] e + c e + c e f. y = ln + c + c g. y = 8 e + c e + c e h. y = ln ) + c + c ln i. y = e + c + c e ] j. y = + + c + c e a. y = 5 ln
14 68 Answers to Selected Eercises b. y = e e + e a. y = e + c e + c e b. y = ln + c + c + c a. u + v + w = 0 u + v + w = 0 v + 6w = e b. e u + cos)v + sin)w = 0 e u sin)v + cos)w = 0 e u cos)v sin)w = tan) c. e u + e u + cos)u + sin)u = 0 e u e u sin)u + cos)u = 0 e u + e u cos)u sin)u = 0 e u e u + sin)u cos)u = 8 sinh) d. u + u + u + u = 0 u u + u u = 0 u + u u = 0 u + u 0 6 u = sin ) Chapter 6a. s for s > 0 6b. s for s > 6c. ] e s 6d. s s e s for s > 0 6e. e s) 6f. e s e s)] 6g. e s s s s s e s 6h. s + ] s e s for s > 0 7a. s for s > 0 7b. 9! 5 s 0 for s > 0 7c. s 7 for s > 7 7d. s + 7 for s > 7 7e. s i7 for s > 0 8a. s + 9 for s > 0 8b. s s + 9 for s > 0 8c. 7 s for s > 0 s 8d. s 9 for s > 8e. s 6 for s > 8f. 6 s 8 s + 7 for s > 0 s 6 8g. s + 8 s + for s > 8h. s s + + s + 6 for s > 0 8i. s 8 s + for s > 0 π 9a. s 5 / 5 π ) for s > 0 9b. s 7 / for s > 0 9c. Ŵ s / for s > 0 ) 8 5 9d. Ŵ s 5 / for s > 0 9e. s e s 0b. e s a. s s ) for s > b. s ) for s > c. 5 s ) + 9 for s > d. s s ) + 9 for s > π e. s ) / for s > f. s e s ) for s > a. piecewise continuous b. piecewise continuous c. piecewise continuous d. piecewise continuous e. not piecewise continuous f. piecewise continuous g. not piecewise continuous h. piecewise continuous i. not piecewise continuous j. piecewise continuous k. piecewise continuous 6a. of eponential order 6b. of eponential order < 0 6c. of eponential order < 6d. not of eponential order 6e. of eponential order 0 7a. The maimum occurs at t = α σ and is M α,σ = ) α α e α. σ
15 Answers to Selected Eercises 69 Chapter 5 a. s + b. s + 6 s s ) c. e s ss + ) d. s + s + 6 s s ) s e. 8 s ) s + ) f. s + 5 s + + s + ) 5 g. s + + e s s s + ) s + 5 h. s + 5s s ) s + 5s + 6 ) i. s 5s s ) s 5s + 6 ) s 6 j. s 5s s s 5s + 6 ) k. s s s + + s s + ) s + 9 l. s + s + + s s + s + ) + 6 s s + ) s + s + ) s m. + s + + s 7 s + ) s 7 ) a. s 9 s + 9 ) 8s b. 5 s + 9 ) c. 6 + d. e. e s C b. Ys) = s + 7) s c. Y0) = s + d. Ys) = s + 5a. 6b. s e αs for s > 0 7. ) s 8b. ln for s > s 8e. ln ) for s > s ωs ) s ) s ] + ω ) 5b. ω s ] + ω ) 5c. ) s arctan 8a. ln + ) for s > 0 s ) s s 8c. ln for s > 8d. s + ln + ) ) s 8f. arctan for s > 0 s Chapter 6 ) a. e 6t b. e t c. t d. t e. sin5t) f. cos πt s + 7) s s ) e s ) for s > 0 a. 6e t b. 6 t c. 8e t d. t πt 6 t e. cos5t) + 5 sin5t) f. step t) a. y) = e 9t b. cost) + sint) 5a. e t + e t 5b. e t e t 5c. et e t 5d. + sint) 5e. 6 et t 6 5f. 8 cost) + e t + e t 5g. e 6t + cost) sint) 5h. cost) + sint) 5i. e t + 5e t e 7t 7 6a. et + e t 6b. t t + sint) 6c. e t + e 7t + e t 7a. t e 7t 7b. 6 et sin6t) 7c. e t cos6s) + ] sin6t) 7d. e t 7e. te t 7f. cost) + sint)]e 6t πt 7g. sint)e 6t 7h. t + t + 8 t ] e t 8a. e t cost) 8b. et t 8c. e t cost) + 7 sint)] 8d. e t cost) 9a. ] + t cost)e t 9b. e t + sin6t) cos6t)] e t 9c. e t te t + e t 9d. t ] cost) 9 ) 6 8 sint) e t 0a. 9 cost)] 0b. e t ] 0c. ] + t )e t 9 Chapter 7. 6 e7t + tet 6 et a. e 5t e t] 6 b. 5 t 5 / c. t / d. e t t ] e. 9 0 t 5 f. sint) t g. sint) t cost)] h. 0 if t <, and e t ) ] if < t i. 0 if t <, and t t + 9t 9 if
16 60 Answers to Selected Eercises < t a. e t e t b. e t ] c. cost)] d. e t sint) cost) ] e. sint) t cost)] 0 5 f. sint) + t cost)] g. 0 if t <, and cost ) if < t t h. e t e d 5a. h) = π 0 sin), yt) = t sin) f t ) d 0 5b. h) = e e ], yt) = t e e ] f t ) d 0 t 5c. h) = e, yt) = e f t ) d 0 5d. h) = e sin), yt) = t e sin) f t ) d 0 5e. h) = t cos)], yt) = cos)] f t ) d a. cost)] 6b. 8 sint)] 6c. e t cost) ] sint) 6d. 8 sint) t cost)] 6e. α 8) α sint) sinαt)] 7a. te t e t + ] 7b. te t e t + t + ] 7c. 9 7 t e t 7d. 6te t e t + e t] 7e. α) α)te t e t + e αt] 6 8a. 6 t sint)] 8b. 8t + cost) ] 56 8c. 6e t cost) sint) ] 8d. t sint) cost)] e. 6a a 6 )] 6 cosat) + a 6 a cost) ] Chapter 8 a. s e 6s ) b. s e 6s + 6 s e 6s a. t ) step t) b. e t ) step t) c. t step t) d. sinπt) step t) e. t ) e 5t ) step t) f. e t 5) cost 5)) step 5 t) { } t if t < 0 if t <. cost )] step t) 5a. 5b. if < t < if < t if < t { } { } { } t if t < sinπt) if t < e t ) if t < 5c. 5d. 5e. if < t 0 if < t e t ) if < t 0 if t < t if < t < 5f. 6a. t ) step t) 6b. + t ) step t) 6 t if < t < 6 0 if 6 < t 6c. t ) step t) 6d. 6t + + t ) step t) ] π 6e. cost 0]) step0 t) 7a. 9 s e 6s ) 7b. ] s e s 7c. s e 6s + 6 s e 6s 7d. s + e 6 s 7e. e s s ) s + s + 6 ] s
17 Answers to Selected Eercises 6 7f. e t 7g. e πt/ 7h. e πt/ s 7i. s + s + s + e πt/6 s + + s ] s + 8a. Fs) = ] e 6s 8b. Fs) = ] + e s ] e s s + s s 8c. Fs) = ] e s + s s + e s 8d. Fs) = π ] + e s s + π 8e. Fs) = s s + 6 ] e s 8f. Fs) = e s e s] s s 8g. Fs) = + e s + e s] 8h. Fs) = π e s + e s] s s + π 8i. Fs) = s e s s + ] e s 8j. Fs) = e s + e s] s s 9a. step n t) 9b. e ns 9c. s s e s] n=0 n=0 0a. t ) rect,) t) + step t) 0b. t ) rect,) +t ] step t) ] 0c. cost ]) rect,) t) + 9 ] cost ]) cost ]) step t) 9 { } 0 if t < 0 if t < a. b. c. e. f. g. h. b. c. d. e. f. t t + 9t 9 if < t t if < t { } sint) if t < π 0 if t < d. e t if < t < sint) if π < t e 6 t e t if < t 0 if t < 7 e 5t e 7 t if < t < e t e 7 t if < t 0 if t < π sint) + π t) cost) if π < t < π π cost) if < t { } t 5/ if t < 5 8 0t + 5t if < t cost) if t < π t π) sint) if π < t < π a. π sint) cost) if π < t ) e s s e s) equivalently, s + e s) e s + e s s e e s) equivalently, s s + e s) or s ) ) s tanh + e s) s e s) equivalently, s s coths) ) s e s + e s s e e s) equivalently, s s + e s) or ) s tanhs) ) ) + e πs s + ) e πs) equivalently, πs s + coth e 6 e s s + ) e s)
18 6 Answers to Selected Eercises a. i) yt + π) yt) = cosπt π) π m a. ii) yt) = π m y 0τ) n cosπt)] with y 0 τ) = { cosπτ) if 0 τ < cosπt) if < τ < Y a. iii) T b. i) yt + π) yt) = π m cos πt π ) b. ii) yt) = sinπτ) cosπτ)] n π m π m sinπt) Y b. iii) T c. i) yt + π) yt) = 0 No resonance!) Y c. ii) yt) = π m sinπt) sinπt)] c. iii) T Chapter 9 a kg meter/sec b..6 kg meter/sec a i. 0 a ii. 0 a iii. 0 b i. 8 b ii. 8 b iii. 8 c i. 0.5 kilogram c ii. kilogram a. 6 b. 0 c. d. e. 9 f. 0 6a. yt) = step t) 6b. yt) = rect,) t) 6c. yt) = t ) step t) 0 if t < 6d. yt) = t if < t < 6e. e t ) stept ) if < t { sint) if t < π 6f. yt) = 6g. cost) step t π ) 0 if t < π 7a. e t + e t ) stept ) 7b. ] e t 7c. e t + ] e t )t stept ) 7d. sint )) step t) 7e. e t 0) e t 0)] step 8 0 t) 7f. 0 7g. e t e 6t] 8 7h. e t ) e 6t )] step 8 t) 7i. t )e t ) step t) 7j. e6t sint) 7k. 9 cost ))] step t) 7l. e t ) e t ) + ] sint )) step t)
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