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凰举廉
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1 - -. act dirntial quation 若 d d d d d 0 act c O.D.E. d d 0 O.D.E. d d 0 c d d d 0 列 c 6 act 念若 c d dc d d 0 d d 0

2 - 量來若 j i F 列兩 0 F 0 k F d r d r d F d d d. Intgrating actor 若 O.D.E. 0 d d O.D.E. 數 I 0 d I d I I I I 0 d d

3 - 若 c d d 0 d d 0 不復便. 列 -dpndnt I d -dpndnt I d -dpndnt I d -dpndnt I d

4 -4 -dpndnt d I -dpndnt d I -dpnd nt d I 4 -dpndnt d I 若 d I O.D.E. 0 d d 0 d d d d d d } { } { d d d d d

5 -5 d 若 d I O.D.E. 0 d d 0 d d d d d d } { } { d d d d d d 若 d I O.D.E. 0 d d 0 d d d d d d } { } { d d d d

6 -6 d [ ] 4 若 I d O.D.E. d d 0 d d d d 0 { } { d d } d d d d d

7 -7 Solv th dirntial quation: 0 d d with 0. 6%95 O.D.E. 0 d d 令 k k c B.C. 0 c Find gnral solution o th quation d d.0%95 北

8 -8 d d 0 d d 令 6 k k c Solv th quation 0 to obtain th solution or as a unction o. 0%95 O.D.E. 0 0 d d 令 k k c

9 -9 Solv 0 d d. 5%95 令 4 k k c 4 5%95 d d 0 d d 令 k k 4 5

10 -0 c 6 Solv sin d d 0 subjct to 0. 0%95 利 sin 令 sin sin c sin k k 7 O.D.E. 0. a Show that th dirntial quation is not act. b Find an intgrating actor o th orm a b. c Find th gnral solution. 4%90

11 - a 0 d d 0 b O.D.E. d d 0 I a b a b a b d d 0 a b a b a b a b a b a b a b a b I a b O.D.E. d d 0 例 a b I c O.D.E. d d 0 I d d 0 d d 0 ln ln ln c ln ln ln c ln ln c c 8 Th intgrating actor or th ollowing dirntial quation is in th orm o σ a b whr a and b ar constants. Find th valus o a and b and us th intgrating actor to ind th gnral solution o th dirntial quation 4. 0%95

12 - d d 4 4 d d 0 σ a b a b a b d d 0 a a b [ b 4 a b b b b ] a a b b 4 b b b a b a b 4 σ 4 d d 0 c 4 9 Solv d d 0. 5%95 北令 4 d I O.D.E. d d 0 d d 0 c

13 - 0 4 sin sin 4 d d 0%95 北令 4 sin sin 4 sin 8 sin 4 sin sin sin sin 4 sin 4 sin 4 sin I d sin 0 4 sin sin 4 d d 0 4 sin sin 4 d d c 4 Solv th dirntial quation 0 sin sin d d 5%95 O.D.E. 0 sin sin d d 0

14 -4 令 sin sin 0 d I O.D.E. d sin sin d 0 d sin sin d 0 sin k sin sin sin c sin k 列 d d 0 95 令 d I O.D.E. d d 0 d d 0 c c

15 -5 令 d I O.D.E. d d 0 d d 0 k k k k c c d d 0 9 d d 0 令 d I O.D.E. d d 0 d d 0

16 -6 ln c

17 -7 4 Choos th corrct answrs ultipl Choics Considr th ordinar dirntial quation: d d A Th quation is a nonact scond ordr ordinar dirntial quation. B Th quation has an -dpndnt intgrating actor. C Th quation has a -dpndnt intgrating actor. D Th quation has an -dpndnt intgrating actor. E All th abov choics ar incorrct. 5%94 O.D.E. d d 0 令 A nonact but irst ordr. B Ys! C o!

18 -8 D o! B α α d. 0 d 參數 α Eact Dirntial Equation 94. Find th valu o b or which th ollowing quation b d d 0 is act and thn solv it using th valu o b. 9 數. d sin d 0 9 精 4. d d. 9 d d 5. d d 9 6. sin d d

19 -9 9. d d Find all unctions with. 0%9 sin. 若 sin 0 90 北. Suppos th unctions and ar continuous and hav continuous irst partial drivativ in th opn rctangl S: a < < b c < < d. I th dirntial quation d d 0 is act thn 0 d. %95. 列 sin ' 0 0%94 4. a Solv that th dirntial orm sin sinh d h d is act. b Solv th dirntial quation sin sinh d h d 0. 0%94 5. Find th gnral solution o d d d 6 d 0 5%94

20 -0 7. I both µ and µ ar intgrating actors or th dirntial quation thn what is? 0 % d sin d %9 0. d d 0 9 北. d d 0 9 d sin. d 49 d. 4 d 4 0%9 d 4. [ ] 6 [ ] 0 d 09 d 5. d sin 0%9. α c. c

21 -. c c 6. sin c 4 7. c 4 8. c 9. c 0. sin c. sin 4.. sin c 4. a sin h sin h act quation b sinh c 5. 4 c 6. c c

22 - 9. c 0. c 4. c 6. sin sin c. ln c c 5. sin c

23 - -. P Q I P d I I Q d c P d P d { Q } d c P d {. P d Q } d c 數 P d d P Q d sin LL du 令數 u d d d du O.D.E. P u Q d

24 -4 P Q I P d I I Q d c O nil P d P d { Q } d c P d { P d Q } d c P d

25 -5 Considr th irst ordr dirntial quation P Q Prov that P d I is an intgrating actor. Find th gnral solution. 9 9 北 90 北 d O.D.E. P Q d I P d d Q d [ P Q ] d d 0 P d [ P d P d P Q ] d [ P d ] d 0 P d P P d Q P d I P d P d d O.D.E. P Q I d d P d P d P d P Q d d P d P d Q d P d P d { Q } d c P P d P d { P d Q } d c P d

26 -6 Lt and b dind on a rgion R o th plan. d Thn µ is an intgrating actor or 0 d d i µ 0 or all in R and µ µ 0 is act on R. d Thror w can gt µ µ. Plas us abov quation to ind th intgrating actor o a d irst-ordr linar dirntial quation P Q. d Find th solution o d d sin 0 tan. 0%95 北冷 µ 來 O il Zill Krszig I 來 d O.D.E. P Q d P d d Q d [ P Q ] d d 0 µ P d [ P d P d P Q ] d [ P d ] d 0 P d P P d Q P d P d P

27 -7 P d µ sin d d tan µ µ sin d c sin d c sin d sin c sin c sin sc csc Solv th initial valu problm d sin 0. 0%95 d d d sin sc tan d d sc d I sc tan sc tan tan d sc tan I tan d sc tan sc d sc tan c c sc tan sc tan I.C. 0 c

28 -8 sc tan sc tan 4 d? 5%95 d d d d d d I I d ln c ln c 5 Plas ind a ral gnral solution or ollowing dirntial quation. show all dtails o our calculations d sin 95 d d sin d d I 4 I I sin d sin d sin c 4 4 c 4 sin

29 -9 6 d Find th particular solution o th quation. d d d d d I 7 I d c c 0%95 北 d sin 0 0 d 0%90 理 sin sin d sin sin d d I sin sin sin I sin d sin d sin sin sin sin c sin c B.C. 0 0 c c sin

30 -0 sin sin 8 Dtrmind th solution o th dirntial quation tan sin Subjct to th initial condition o 0. ot that tan d ln 96 北冷 95 北 9 9 北冷 90 I tan d ln ln I I sin d sin d sin d c c B.C. 0 c c 9 [sin ] [sin ] sin ot sc d tan 59 北 sin sin sin sin sin sin

31 - tan I tan d sin d ln ln I I d sc d tan c tan c sin c 0 intgral quation t dt 59 北 t t dt t I d I I d d c 9 t 6 dt B.C. t 6 c

32 - 6 6 c 6 c 6 6 d P Q d sin LL du 令數 u d du O.D.E. P u Q d d d sin 9 令 u sin u O.D.E.u u d I I u d c u c sin c

33 - tan 9 令 u tan u O.D.E. u u d I c I u d c u c c tan tan h 4 d sinh d 0 0%9 北 d sinh h 4 d 令 u h u sinh d d O.D.E. u u 4 u u 4 d I

34 -4 4 I u I 4 d u 4 d c c u c h d P Q d I P d I I Q d c 4 Solv d d d d 0 d d d d d d d I I d d c c

35 -5 5 sin 0%9 d d d d sin sin sin d d d I I sin d sin c sin c Find th gnral solution o th quation. d d 6 d d d d 0%95 數 90 d Ι Ι d c c

36 料 ; 9 d 5. sin 0 d 9 6. t 6 t t t > 0. 9 北 7. sinτ dτ is th solution o ' sin τ on th positiv -ais A 流 5 A A 行度流 5 A A 量 A 量? A 不 A A 5%95 9. sinθ dθ s θ θ d 0 5%9 0. Find an intgrating actor o th dirntial quation 6 0%95 d sin π. 0 d 90 d. d 0%94

37 北 d 4. Find th gnral solution o. 5%94 d ln 5. A bod o mass m is thrown vrticall into th air with an initial vlocit v 0. I th bod ncountrs an air rsistanc proportional to its vlocit ind a th vlocit o th bod at an tim t and b th tim at which th bod rachs its maimum hight. 0%94 6. A radioactiv substanc with dca constant k is producd at a constant rat o a units o mass pr unit tim. Assuming that Q 0 Q0 ind th mass Q t o th substanc prsnt at tim t. 0% Find lim Q t. 5% 94 t c sin c c t 6 sin

38 -8 7. sinτ dτ τ 8. t 400 kg 9. θ s c s 0.. I tan sin sin. d c. ln c 4. c ln ln mg mg t m 5. av t V0 k k mg m b t ln mg k k V0 k 6. Q t Q 0 a Q t k a k kt a k k

39 -9 - BrnoulliRiccati. Brnoulli quation d d α P Q α 0. Riccati quation d P Q R d Brnoulli d d 令 u P Q α α d α P Q d α du d α 0 α α d d du P u Q α d

40 -40 Find th gnral solution o this ollowing dirntial quation: 5 5%95 北 5 O.D.E du 5 令 u 4 d 4 du d u du 4u 4 d 4d I I u d c u c c 4 Solv d d. 5%95 d d 令 u du d d d du du u u d d d I

41 -4 I u d c u c c d Find th gnral solution o th dirntial quation. d du 令 u d du u d du d I u d I u d c u c c 4 0%95 Solv 5%95 北

42 -4 du 令 u d du d du u u d d I 5 I u d c u c c Considr th dirntial quation > 0. a Transorm th abov dirntial quation into irst-ordr linar dirntial quation. b Find th gnral solution. 0%95 a > 0 du 令 u d du du u u d d b d I

43 -4 I u d c u c 6 若 90 m b Brnoulli 令 u u O.D.E. u u d I c Ι u d c u B.C. c c 7 Th Brnoulli quation is in th orm o P α Q R in which α is a constant. Plas ind th intgrating actor µ. Hint: tr µ α 0%9 北

44 -44 α R Q d d P [ ] 0 d R Q d P α α µ [ ] 0 d R Q d P α α [ ] [ ] R Q P α α α α α Q P Q d P d α d P Q P P d α d P Q P ln α d p Q P α d P Q P α d P Q P α α µ

45 -45 Riccati P Q R 若讀行令 b b a a P Q R 令 z z P Q R z P z Q R z P R z Rz z 若讀行令 b b a a P Q R 令 O il z P Q R -5

46 -46 8 Solv th ollowing dirntial quations: 0%95 Riccati b b b 令 a ab a a a b 令 z z z z z z z z z z z z z 令 u z u z z d I I u d ln c c u ln b u u u u

47 -47 z ln c 9 d 0. 9 d d d d d Riccati b d 令 a O.D.E. d b b b a ab a a a a a b 0 令 z z z O.D.E. d d z z z z z z z z Brnoulli z z z z 令 u z u z z O.D.E. u u d I I u d c

48 -48 u c z u c B.C. 0 0 c c c 讀更 0 0 0%9 O.D.E. b 令 a O.D.E. b b b a b a a b b a ab - a b 0 令 z z z O.D.E. z z z z z z z z z z Brnoulli

49 -49 z z z z 令 u z u z z O.D.E. u u d I I u d c c u z u c 讀更 d d 令 0%90 b d a O.D.E. d b b ab a a b b ab a a b b a b 令 z z z 4 z 4z 4 z z z Brnoulli

50 -50 z z z 令 u z u z z O.D.E. u u u u d I d d d I u c u d c z u d c d c z d c

51 -5. 列 4 ; 4. 0%9 北. Solv d d 9 6 0%9 d 5. Solv th ollowing dirntial quation. d hint: a Brnoulli dirntial quation 0% ; ; 0%94 d.? d 0%94 d.? d 0%94. Dtrmin th gnral solution o th dirntial quation ' which has a particular solution Y b inspction.

52 -5 / / C d / 4 / C d C / C / / d / d 5 7 / 6 / C d 8 C d C d C d 9 0. C d C d 7% c. c c 5 c

53 -5 7. c 8. ln c. c. d c.

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