Differential equations computing and modeling and differential equations and boundary value problems by peter.martinson602

Differential Equations Computing and Modeling and Differential Equations and Boundary Value Problems Computing and Modeling 5th Edition Edwards Solutions Manual Visit to Download in Full: https://testbankdeal.com/download/differential-equations-co mputing-and-modeling-and-differential-equations-and-boundary-value-problems-com puting-and-modeling-5th-edition-edwards-solutions-manual/

LINEAR EQUATIONS OF HIGHER ORDER

SECTION3.1

INTRODUCTION: SECOND-ORDER LINEAR EQUATIONS

Inthissectionthecentralideasofthetheoryoflineardifferentialequationsareintroducedand illustratedconcretelyinthecontextof second-order equations.Thesekeyconceptsincludesuperpositionofsolutions(Theorem1),existenceanduniquenessofsolutions(Theorem2),linear independence,theWronskian(Theorem3),andgeneralsolutions(Theorem4).Thisdiscussion ofsecond-orderequationsservesaspreparationforthetreatmentof nthorderlinearequationsin Section3.2.Althoughtheconceptsinthissectionmayseemsomewhatabstracttostudents,the problemssetisquitetangibleandlargelycomputational.

IneachofProblems1–16theverificationthat1 y and2 y satisfythegivendifferentialequationis aroutinematter.AsinExample2,wethenimposethegiveninitialconditionsonthegeneral solution1122

167

CHAPTER3

y cycy

c and2 c 1. Impositionoftheinitialconditions 00 y  , 05 y   onthegeneralsolution  12 x x yxcece  yieldsthetwoequations120 cc ,125 cc withsolution15 2 c  , 2 5 2 c  .Hencethedesiredparticularsolutionis   5 2 x x y xee  2. Impositionoftheinitialconditions 01 y  , 015 y   onthegeneralsolution  33 12 x x yxcece  yieldsthetwoequations121 cc ,123315 cc ,withsolution 12c  ,23 c  .Hencethedesiredparticularsolutionis  3323 x x yxee  . 3. Impositionoftheinitialconditions 03 y  , 08 y   onthegeneralsolution  12 cos2sin2 yxcxcx  yieldsthetwoequations13 c  ,228 c  withsolution13 c  , 24c  .Hencethedesiredparticularsolutionis  3cos24sin2 yxxx  . 4. Impositionoftheinitialconditions 010 y  , 010 y   onthegeneralsolution  12 cos5sin5 yxcxcx  yieldsthetwoequations110 c  ,2510 c  withsolution 13c  ,24 c  .Hencethedesiredparticularsolutionis  10cos52sin5 yxxx  Differential Equations Computing and Modeling and Differential Equations and Boundary Value Visit TestBankDeal.com to get complete for all chapters

.Thisyieldstwolinearequationsthatdeterminethevaluesoftheconstants1

168 INTRODUCTION:SECOND-ORDERLINEAREQUATIONS Copyright©2015PearsonEducation,Inc. 5. Impositionoftheinitialconditions 01 y  , 00 y   onthegeneralsolution  2 12 x x yxcece  yieldsthetwoequations121 cc ,1220cc withsolution 12c  ,21 c  .Hencethedesiredparticularsolutionis  22 x x yxee  . 6. Impositionoftheinitialconditions 07 y  , 01 y   onthegeneralsolution  23 12 x x yxcece  yieldsthetwoequations127 cc ,12231 cc withsolution 14c  ,23 c  .Hencethedesiredparticularsolutionis  2343 x x yxee  7. Impositionoftheinitialconditions 02 y  , 08 y   onthegeneralsolution  12 x yxcce  yieldsthetwoequations122 cc ,28 c  withsolution16 c  , 28c  .Hencethedesiredparticularsolutionis  68 x yxe  . 8. Impositionoftheinitialconditions 04 y  , 02 y   onthegeneralsolution 3 ()12 x y xcce  yieldsthetwoequations124 cc ,232 c  withsolution114 3 c  , 2 2 3 c  .Hencethedesiredparticularsolutionis  13 142 3 x y xe  . 9. Impositionoftheinitialconditions 02 y  , 01 y   onthegeneralsolution  12 x x yxcecxe  yieldsthetwoequations12 c  ,121 cc  withsolution 12c  21c  .Hencethedesiredparticularsolutionis  2 x x yxexe  . 10. Impositionoftheinitialconditions 03 y  , 013 y   onthegeneralsoution  55 12 x x yxcecxe  yieldsthetwoequations13 c  ,12513 cc withsolution13 c  , 22c  .Hencethedesiredparticularsolutionis  5532 x x yxexe  . 11. Impositionoftheinitialconditions(0)0 y  , 05 y   onthegeneralsolution  12cossin xx yxcexcex  yieldsthetwoequations10 c  ,125 cc withsolution 10c  ,25 c  .Hencethedesiredparticularsolutionis  5sin x yxex  12. Impositionoftheinitialconditions 02 y  , 00 y   onthegeneralsolution  33 12cos2sin2 xx yxcexcex  yieldsthetwoequations12 c  ,12325 cc  with solution12 c  ,23 c  .Hencethedesiredparticularsolutionis  32cos23sin2 x yxexx  .

Section3.1 169 Copyright©2015PearsonEducation,Inc. 13. Impositionoftheinitialconditions 13 y  , 11 y   onthegeneralsolution  2 12 yxcxcx  yieldsthetwoequations123 cc ,1221cc withsolution15 c  , 22c  .Hencethedesiredparticularsolutionis  522yxxx  . 14. Impositionoftheinitialconditions 210 y  , 215 y   onthegeneralsolution  23 12 yxcxcx  yieldsthetwoequations2 1 410 8 c c  , 2 1 3 415 16 c c  withsolution 13c  ,216 c  .Hencethedesiredparticularsolutionis  2 3 16 3 yxx x  . 15. Impositionoftheinitialconditions 17 y  , 12 y   onthegeneralsolution  12ln yxcxcxx  yieldsthetwoequations17 c  ,122 cc withsolution17 c  , 25c  .Hencethedesiredparticularsolutionis  75ln yxxxx  . 16. Impositionoftheinitialconditions 12 y  , 13 y   onthegeneralsolution  12 coslnsinln yxcxcx  yieldsthetwoequations12 c  ,23 c  .Hencethedesiredparticularsolutionis  2cosln3sinln yxxx  17. If c y x  ,then  2 2 222 1 0 cc cc yy xxx   unlesseither0 c  or1 c  . 18. If3 y cx  ,then3244 666 yycxcxcxx   unless21 c  19. If1yx  ,then   2 321232 210 424 xxx yyyx     . 20. Linearlydependent,because    22 cossin f xxxgx   . 21. Linearlyindependent,because32 x xx  if0 x  ,whereas32 x xx  if0 x  22. Linearlyindependent,because 11 x cx wouldrequirethat1 c  with0 x  ,but 0 c  with1 x  .Thusthereisnosuchconstant c. 23. Linearlyindependent,because  f xgx  if0 x  ,whereas  f xgx  if0 x  . 24. Linearlydependent,because  2 gxfx  .

25.  sin x f xex  and  cos x gxex  arelinearlyindependent,because  f xkgx  wouldimplythatsincos x kx  ,whereassin x andcos x arelinearlyindependent,as notedinExample3.

26. Toseethat  f x and gx arelinearlyindependent,assumethat

 ,and thensubstituteboth0 x  and

27. Theoperatornotationusedelsewhereinthischapterisconvenienthere.Let   Ly denote y pyqy

28. If

with1 x  .

(b) For0 x  ,  12,0Wyy  because21 y y  .For0 x  ,  33 1222 ,0 33 xx Wyy xx 

because21 y y  .At0 x  ,2 y hasleft-andright-handderivativesbothequaltozero, sothat  12 00 ,0 00 Wyy  onceagain.Thus  12 , Wyy isidenticallyzero.

Thefactthat  12,0Wyy  everywheredoesnotcontradictTheorem3,becausewhenthe givenequationiswrittenintherequiredform,namely2 33 0 yyy xx  ,thecoefficientfunctions  3 px x  and  2 3 qx x  arenotcontinuousat0 x  .

31.  12,2 Wyyx  vanishesat0 x  ,whereasif1 y and2 y were(linearlyindependent) solutionsofanequation0 ypyqy  with p and q bothcontinuousonanopeninterval I containing0 x  ,thenTheorem3wouldimplythat0 W  on I.

32. (a) Because1212 Wyyyy   ,wehave

170

INTRODUCTION:SECOND-ORDERLINEAREQUATIONS



 

f xcgx

 .Then   0 c Ly  and p Lyf    ,so0 cp Lyyff  .



 ,then  12sincos yxcxcx   ,sotheinitialconditions  001yy   yield12 c  ,21 c  .Hence  12cossin yxxx  . 29.

 4 px x  and  2 6 qx x  arenotcontinuousat0 x  . 30. (a) 3 1y x  and3 2 yx  arelinearlyindependentbecause33 x cx  wouldrequirethat 1 c  with1 x  ,but1 c 

12 1cossin yxcxcx

Thereisnocontradiction,becauseifthegivendifferentialequationisdividedby

gettheforminEquation(8)inthetext,thentheresultingfunctions

InProblems33–42wegivethecharacteristicequation,itsroots,andthecorrespondinggeneral solution.

Section3.1 171 Copyright©2015PearsonEducation,Inc. 12 dW AAyy dx   121212 yyyyyy      1221 122211 1212 12,, yAyyAy y ByCyyByCy Byyyy BWyy            andthus   dW A xBxWx dx  . (b) Thedifferentialequationfor Wx foundin a canberewrittenas    0 Bx dW Wx dxAx  ,since Ax isneverzero.Wecansolvethisequationasalinear first-orderequation:Multiplyingbytheintegratingfactor    exp Bx x dx Ax       gives        expexp0 BxBxBx dW dxdxWx AxdxAxAx     , or    exp0 Bx d dxWx dxAx          , or    exp Bx dxWxK Ax      , where K isaconstant.Finallywefind   exp Bx WKdx Ax     ,asdesired.

33. 2320rr ;1,2 r  ;  2 12 x x y xcece  34. 22150rr ;3,5 r  ;  53 12 x x y xcece  35. 250rr ;0,5 r  ;  5 12 x y xcce 

InProblems43–48wefirstwriteandsimplifytheequationwiththeindicatedcharacteristic roots,andthenwritethecorrespondingdifferentialequation.

172

INTRODUCTION:SECOND-ORDERLINEAREQUATIONS

36. 2 230 rr ; 3 0, 2 r  ;  32 12 x yxcce  37. 2 210 rr ; 1 1, 2 r  ;  2 12 x x y xcece  38. 2 4830 rr ; 13 , 22 r  ;  232 12 x x yxcece  39. 2 4410 rr ; 1 2 r  (repeated);  2 12 x y xccxe  40. 2 91240 rr ; 2 3 r  (repeated);  23 12 x y xccxe  41. 2 67200 rr ; 45 , 32 r  ;  4352 12 x x y xcece  42. 2 35120 rr ; 43 , 75 r  ;  4735 12 x x y xcece 

43.  2 010100rrrr  ;100 yy 44.  2 10101000rrr  ; 1000 y y   45.  2 1010201000rrrr  ;201000 yyy  46.  2 1010011010000rrrr  ; 11010000yyy  47.  2 000rrr  ;0 y   48.   2 1212210rrrr    ; 20yyy   49. Thesolutioncurvewith 01 y  , 06 y   is  872 x x yxee  .Wefindthat  0 yx   when7 ln 4 x  ,sothat 4 7 x e  and216 49 x e  .Itfollowsthat716 ln 47 y     ,

Section3.1 173 Copyright©2015PearsonEducation,Inc. sothehighpointonthecurveis716ln,(0.56,2.29) 47     ,whichlooksconsistentwith Fig.3.1.6. 50. Thetwosolutioncurvessatisfying  0 ya  and  0 yb  ,aswellas 01 y   ,aregivenby   2 2 211 211. x x x x y aeae y bebe   Subtraction,followedbydivisionby ab ,gives2 2 x x ee  ,soitfollowsthat ln2 x  .Nowsubstitutionineitherformulagives2 y  ,sothecommonpointofintersectionis  ln2,2 51. (a) Thesubstitutionlnvx  gives 1 dydydvdy y dxdvdxxdv   . Thenanotherdifferentiationusingthechainandproductrulesgives 2 2 2 2 2 222 1 11 11 11 dy y dx ddy dxdx ddy dxxdv dyddy xdvxdxdv dyddydv x dvxdvdvdx dydy x dvxdv                   Substitutionoftheseexpressionsfor y  and y  intoEq.(21)inthetextthenyieldsimmediatelythedesiredEq.(23):  2 20dydy abacy dvdv  .

  12 1212 121212 rr rvrvrr vv y xcececececxcx 

(b) Iftheroots1 r and2 r ofthecharacteristicequationofEq.(23)arerealanddistinct, thenageneralsolutionoftheoriginalEulerequationis

SECTION3.2

GENERAL SOLUTIONS OF LINEAR EQUATIONS

StudentsshouldcheckeachofTheorems1through4inthissectiontoseethat,inthecase2 n  , itreducestothecorrespondingtheoreminSection3.1.Similarly,thecomputationalproblems forthissectionlargelyparallelthosefortheprevioussection.BytheendofSection3.2students shouldunderstandthat,althoughwedonotprovetheexistence-uniquenesstheoremnow,itprovidesthebasisforeverythingwedowithlineardifferentialequations.

ThelinearcombinationslistedinProblems1–6werediscovered“byinspection”—thatis,bytrial anderror.

174 INTRODUCTION:SECOND-ORDERLINEAREQUATIONS Copyright©2015PearsonEducation,Inc. 52. Thesubstitutionlnvx  yieldstheconvertedequation 2 20 dy y dv  ,whosecharacteristic equation210 r  hasroots11 r  and21 r  .Because v ex  ,thecorrespondinggeneralsolutionis2 121 vv c ycececx x  53. Thesubstitutionlnvx  yieldstheconvertedequation 2 2120dydy y dvdv  ,whosecharacteristicequation2120 rr hasroots14 r  and23 r  .Because v ex  ,thecorrespondinggeneralsolutionis4343 1212 vv y cececxcx  54. Thesubstitutionlnvx  yieldstheconvertedequation 2 2 4430 dydy y dvdv  ,whose characteristicequation24430 rr hasroots13 2 r  and21 2 r  .Because v ex  , thecorrespondinggeneralsolutionis3/2/23/21/2 1212 vv y cececxcx  55. Thesubstitutionlnvx  yieldstheconvertedequation 2 20 dy dv  ,whosecharacteristic equation20 r  hasrepeatedroots12,0rr  .Becauselnvx  ,thecorrespondinggeneral solutionis1212ln y ccvccx . 56. Thesubstitutionlnvx  yieldstheconvertedequation 2 2440dydy y dvdv  ,whosecharacteristicequation2440 rr hasroots12,2rr  .Because v ex  ,thecorresponding generalsolutionis  222 1212ln vv y cecvexccv  .

1.  5822 231580 23 xxxx  

forall x.

2.  22 45523110150 xx  forall x

3. 100sin00 x xe  forall x. 4. 221717 1172sin3cos0 23 xx 

  forall x,because22sincos1 xx .

5.  2 11734cos17cos20 xx  forall x,because22cos1cos2 x x  .

eee Weeee eee

23 236 23 232 49

xxx x xxx xxx



 22

 714 x Wxexx

Section3.2 175

 





isneverzero. 9.

cossin x

Wexxe

isneverzero. 10.

isnonzerofor0

6. 11cosh1sinh0 x exx  ,because  cosh1 2 x x x ee and 11.

1 sinh 2 x x x ee . 7. 12 0122 002 x Wxe

xx Wx isnonzeroeverywhere. isnonzeroif0

8. 12.



x  .



x 

 2222 2cosln2sinln2 Wxxxx    isnonzerofor0 x  .



IneachofProblems13-20wefirstformthegeneralsolution

112233 y xcyxcyxcyx  ,thencalculate  y x  and  y x  ,andfinallyimposethe giveninitialconditionstodeterminethevaluesofthecoefficients123 ,, ccc .

176

Copyright©2015PearsonEducation,Inc. 13. Impositionoftheinitialconditions 01 y  , 02 y   , 00 y   onthegeneralsolution  2 123 x xx y xcecece  yieldsthethreeequations 1231231231,22,40ccccccccc  , withsolution14 3 c  ,20 c  ,3 1 3 c  .Hencethedesiredparticularsolutionisgivenby  12 4 3 x x yxee  14. Impositionoftheinitialconditions 00 y  , 00 y   , 03 y   onthegeneralsolution  23 123 x xx y xcecece  yieldsthethreeequations 123123123 1,232,490ccccccccc  , withsolution13 2 c  ,23 c  ,3 3 2 c  .Hencethedesiredparticularsolutionisgivenby  2333 3 22 x xx y xeee  15. Impositionoftheinitialconditions 02 y  , 00 y   , 00 y   onthegeneralsolution  2 123 x xx y xcecxecxe  yieldsthethreeequations 1121232,0,220cccccc  , withsolution12 c  ,22 c  ,31 c  .Hencethedesiredparticularsolutionisgivenby   222 x yxxxe  16. Impositionoftheinitialconditions 01 y  , 04 y   , 00 y   onthegeneralsolution  22 123 x xx y xcececxe  yieldsthethreeequations 121231231,24,440cccccccc  withsolution112 c  ,213 c  ,310 c  .Hencethedesiredparticularsolutionisgiven by  22121310 x xx y xeexe  . 17. Impositionoftheinitialconditions 03 y  , 01 y   , 02 y   onthegeneralsolution  123cos3sin3 y xccxcx  yieldsthethreeequations 13323,31,92cccc 

GENERALSOLUTIONSOFLINEAREQUATIONS

1122 cpp y xyxyxcyxcyxyx  , thencalculate  y x  ,andfinallyimposethegiveninitialconditionstodeterminethevaluesof thecoefficients1 c and2 c .

21. Impositionoftheinitialconditions 02 y  , 02 y   onthegeneralsolution

 12cossin3 y xcxcxx  yieldsthetwoequations12 c  ,232 c  withsolution

12c  ,25 c  .Hencethedesiredparticularsolutionisgivenby

 2cos5sin3 y xxxx 

Section3.2 177 Copyright©2015PearsonEducation,Inc. withsolution129 9 c  ,2 2 9 c  ,3 1 3 c  .Hencethedesiredparticularsolutionisgiven by  2921cos3sin3 993 yxxx  . 18. Impositionoftheinitialconditions 01 y  , 00 y   , 00 y   onthegeneralsolution  123cossin x y xeccxcx  yieldsthethreeequations 12123131,0,20ccccccc  withsolution12 c  ,21 c  ,31 c  .Hencethedesiredparticularsolutionisgivenby  2cossin x y xexx  . 19. Impositionoftheinitialconditions 16 y  , 114 y   , 122 y   onthegeneralsolution  23 123 y xcxcxcx  yieldsthethreeequations 123123236,2314,2622cccccccc  , withsolution11 c  ,22 c  ,33 c  .Hencethedesiredparticularsolutionisgivenby  2323 y xxxx  . 20. Impositionoftheinitialconditions 11 y  , 15 y   , 111 y   onthegeneralsolution  22 123ln y xcxcxcxx  yieldsthethreeequations 12123231,25,6511ccccccc  , withsolution12 c  ,21 c  ,31 c  .Hencethedesiredparticularsolutionisgivenby  22 2ln y xxxxx  . IneachofProblems21-24wefirstformthegeneralsolution 

GENERALSOLUTIONSOFLINEAREQUATIONS

22. Impositionoftheinitialconditions 00 y  , 010 y   onthegeneralsolution

 22 123 xx yxcece yieldsthetwoequations1230 cc ,122210 cc withsolution14 c  ,21 c  .Hencethedesiredparticularsolutionisgivenby

 22 43 xx yxee

23. Impositionoftheinitialconditions 03 y  , 011 y   onthegeneralsolution

 3 122 xx yxcece yieldsthetwoequations1223 cc ,12311cc  withsolution11 c  ,24 c  .Hencethedesiredparticularsolutionisgivenby

 3 42xx yxee .

24. Impositionoftheinitialconditions 04 y  , 08 y   onthegeneralsolution

 12coscos1 xx y xcexcexx  yieldsthetwoequations114 c  ,1218 cc withsolution13 c  ,24 c  .Hencethedesiredparticularsolutionisgivenby 

178

3cos4sin1 x y xexxx  . 25.   1212 LyLyyLyLyfg  26. (a) 12y  and23 y x  (b) 1223 y yyx 27. Theequations 2 12323310,20,20ccxcxccxc  (thelattertwoobtainedbysuccessivedifferentiationofthefirstone)evidentlyimplythat 1230 ccc . 28. Ifwedifferentiatetheequation20120 n n ccxcxcx  repeatedly, n timesinsuccession,theresultisthesystem 2 012 1 12 1 0 20 (1)!!0 !0 n n n n nn n ccxcxcx ccxncx ncncx nc        of1 n  equationsinthe1 n  coefficients012,,,, n cccc  .Thelastequationimpliesthat 0 n c  ,whereupontheprecedingequationgives10 n c  ,andsoforth.Thusitfollows thatallofthecoefficientsmustvanish.

Section3.2 179 Copyright©2015PearsonEducation,Inc. 29. If010 rxrxnrx n cecxecxe  ,thendivisionby rx e yields010 n n ccxcx  ,so theresultofProblem28applies. 30. Whentheequation2220 xyxyy isrewritteninstandardform 2 22 0 yyy xx     , weseethatthecoefficientfunctions1 2 p x  and  22 2 px x  arenotcontinuousat 0 x  .ThusthehypothesesofTheorem3arenotsatisfied. 31. (a) Substitutionof x a  inthedifferentialequationgives  y apyaqa   (b) If 01 y  and 00 y   ,thentheequation250 yyy   impliesthat  020505yyy  . 32. Letthefunctions12,,, n y yy  bechosenasindicated.Thenevaluationat x a  ofthe 1st k derivativeoftheequation11220 cycycynn yields0 ck  .Thus 120 n ccc ,sothefunctionsarelinearlyindependent. 33. Thisfollowsfromthefactthat  222 111 abcbacbca abc  whena,b,andcaredistinct,whichcanbeverifiedbyexpandingbothsidesoftheequation. 34.  121 ,,,exp n ni i WfffVrx       ,andneither V nor exp1 n i i rx       vanishes. 36. If1 y vy  ,thensubstitutionofthederivatives11 y vyvy   ,111 2 y vyvyvy   inthe differentialequation0 ypyqy  gives       111111 20vyvyvypvyvyqvy   , or   11111120vypyqyvyvypvy   . Butthetermswithinbracketsvanishbecause1 y isasolution,andthisleavestheequation  11120yvypyv   .

tionandsimplify,wegetthe

180 GENERALSOLUTIONSOFLINEAREQUATIONS Copyright©2015PearsonEducation,Inc. Wecansolvethisbyseparatingvariablesandintegrating:1 1 2 vy p vy    leadsto  1 ln2lnln vypxdxC   , or   2 1 pxdx C vxe y    , or   2 1 pxdx e vxCdxK y      . With1 C  and0 K  thisgivesthesecondsolution   212 1 pxdx e y xyxdx y      37. Whenwesubstitute3 y vx  inthegivendifferentialequationandsimplify,wegetthe separableequation0 xvv ,whichwewriteas1 v vx    .Integratinggives lnlnlnvxA   ,andthensolvingfor v  leadsto A v x   ,orfinally  ln vxAxB  With1 A  and0 B  weget  ln vxx  ,andthus  3 2ln y xxx  . 38. Whenwesubstitute3 y vx 

xvv ,whichwewriteas7 v vx    .Integratinggives ln7lnlnvxA   ,andthensolvingfor v  leadsto7 A v x   ,orfinally  66 A vxB x  .With6 A  and0 B  weget  6 1 vx x  ,andhence  23 1 yx x  . 39. Whenwesubstitute2 x y ve  inthegivendifferentialequa

allygetthesimpleequation0 v   ,withgeneralsolution  vxAxB  .With1 A  and0 B  weget  vxx  ,andhence  2 2 x y xxe  40. Whenwesubstitute y vx  inthegivendifferentialequa

separableequation0 vv   ,whichwewriteas1 v v    .Integratinggives

inthegivendifferentialequa

separableequation70

tionandsimplify,weeventu-

tionandsimplify,wegetthe

Section3.2 181 Copyright©2015PearsonEducation,Inc. lnlnvxA   ,andthensolvingfor v  leadsto x vAe   ,orfinally  x vxAeB  . With1 A  and0 B  weget  x vxe  ,andhence  2 x y xxe  41. Whenwesubstitute x y ve  inthegivendifferentialequationandsimplify,wegetthe separableequation 10 xvxv   ,whichwewriteas1 1 11 vx vxx     .Integratinggives  lnln1ln vxxA   ,andthensolvingfor v  leadsto  1 x vAxe   ,orfinally  12xx vxAxedxAxeB   .With1 A  and0 B  weget  2 x vxxe  ,andhence  22 y xx  . 42. Whenwesubstitute y vx  inthegivendifferentialequationandsimplify,wegetthe separableequation 212 x xvv   ,whichwewriteas  2 2211 111 v vxxx xx     Integratinggives  ln2lnln1ln1ln vxxxA   , andthensolvingfor v  leadsto  2 22 11 1 Ax vA xx     ,orfinally  1 vxAxB x     .With1 A  and0 B  weget  1 vxx x  ,andhence  2 21yxx 43. Whenwesubstitute y vx  inthegivendifferentialequationandsimplify,wegetthe separableequation  22 124 x xvxv   ,whichwewriteusingthemethodofpartial fractionsas  2 2 24211 111 vx vxxx xx     . Integratinggives  ln2lnln1ln1ln vxxxA   , andthensolvingfor v  leadsto    222 111 12121 A vA x xx xx          , orfinally

SECTION3.3

HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS

Thisisapurelycomputationalsectiondevotedtothesinglemostwidelyapplicabletypeofhigherorderdifferentialequations—linearoneswithconstantcoefficients.InProblems1–20,we firstwritethecharacteristicequationandlistitsroots,thengivethecorrespondinggeneralsolutionofthegivendifferentialequation.Explanatorycommentsareincludedonlywhenthesolutionofthecharacteristicequationisnotroutine.

182 GENERALSOLUTIONSOFLINEAREQUATIONS Copyright©2015PearsonEducation,Inc.    111 ln1ln1 22 vxAxxB x     . With1 A  and0 B  weget    111 ln1ln1 22 vxxx x  ,andhence  2 1ln1 21 x x yx x   44. Whenwesubstitute1/2cos y vxx  inthegivendifferentialequationandsimplify,we eventuallygettheseparableequation  cos2sin x vxv   ,whichwewriteas 2sin cos vx vx    .Integratinggives 2 ln2lncoslnlnsecln vxAxA   , andthensolvingfor v  leadsto2 sec vAx   ,orfinally  tan vxAxB  .With1 A  and0 B  weget  tan vxx  ,andhence  1/21/2 2tancossin yxxxxxx 

1.  24220 rrr ;2,2 r  ;  22 12 x x y xcece  2.  2 23230 rrrr ; 3 0,2 r  ;  32 12 x y xcce  3.  2310520 rrrr ;5,2 r  ;  25 12 x x y xcece  4.  2 2732130 rrrr ; 1 2,3 r  ;  23 12 x x y xcece  5.  226930rrr ;3 r  (repeated);  33 12 x x y xcecxe  6. 2550rr ; 55 2 r   ;  5555 22 12 x x yxcece  

Section3.3 183 Copyright©2015PearsonEducation,Inc. 7.  22 4129230 rrr ; 3 2 r  (repeated);  3232 12 x x y xcecxe  8. 26130rr ; 616 32 2 ri   ;  3 12 cos2sin2 x y xecxcx  9. 28250rr ; 836 43 2 ri   ;  4 12 cos3sin3 x y xecxcx  10.  433 53530 rrrr ; 3 0,0,0, 5 r  ;  235 1234 x yxccxcxce  11.  4322281640rrrrr ;0,0,4,4 r  ;  44 1234 x x y xccxcecxe  12.  43233310rrrrrr ;0,1,1,1 r  ;  2 1234 x xx y xccecxecxe  13.  322 9124320 rrrrr ; 22 0,, 33 r  ;  2323 123 x x yxccecxe  14. 422234140rrrr ;1,1,2 ri  ;  1234cos2sin2 xx y xcececxcx  15.   422222 48164220 rrrrr  ;2,2,2,2 r  ;  2222 1234 x xxx y xcecxececxe  16.  4222188190rrr ;3,3 rii  ;  1234cos3sin3 yxccxxccxx  17. 4222 611421340 rrrr  ; 2 , 23 ii r  ;  1234 22cossincossin 2233 x xxx yxcccc  18. 42216440rrr ;2,2,2 ri  ;  22 1234cos2sin2 xx yxcececxcx  19. Factoringbygroupinggives   32222 111110rrrrrrrr  ; 1,1,1 r  ;  123 x xx y xcececxe 

184 HOMOGENEOUSEQUATIONSWITHCONSTANTCOEFFICIENTS Copyright©2015PearsonEducation,Inc. 20.  43222 232110rrrrrr  ; 1313 , 22 ii ;    22 1234 33cossin 22 xx y xeccxxeccxx    21. Impositionoftheinitialconditions 07 y  , 011 y   onthegeneralsolution  3 12 x x y xcece  yieldsthetwoequations127 cc ,12311cc withsolution 15c  ,22 c  .Hencethedesiredparticularsolutionis  523 x x y xee  22. Impositionoftheinitialconditions 03 y  , 04 y   onthegeneralsolution  3 12 cossin 33 x x x yxecc    yieldsthetwoequations13 c  , 124 33 cc  with solution13 c  ,253 c  .Hencethedesiredparticularsolutionis  33cos53sin 33 x x x yxe    23. Impositionoftheinitialconditions 03 y  , 01 y   onthegeneralsolution  3 12 cos4sin4 x y xecxcx  yieldsthetwoequations13 c  ,12341 cc withsolution13 c  ,22 c  .Hencethedesiredparticularsolutionis  33cos42sin4 x y xexx  . 24. Impositionoftheinitialconditions 01 y  , 01 y   , 03 y   onthegeneralsolution  22 123 x x y xccece  yieldsthethreeequations 33 123221,21,43 24 cc ccccc  , withsolution17 2 c  ,2 1 2 c  ,34 c  .Hencethedesiredparticularsolutionis  7122 4 22 x x y xee  . 25. Impositionoftheinitialconditions 01 y  , 00 y   , 01 y   onthegeneralsolution  23 123 x yxccxce  yieldsthethreeequations 33 132 24 1,0,1 39 cc ccc ,

Section3.3 185 Copyright©2015PearsonEducation,Inc. withsolution113 4 c  ,2 3 2 c  ,3 9 4 c  .Hencethedesiredparticularsolutionis  133923 424 x yxxe  . 26. Impositionoftheinitialconditions 01 y  , 01 y   , 03 y   onthegeneralsolution  55 123 x x y xccecxe  yieldsthethreeequations 1223233,54,25105cccccc  , withsolution124 5 c  ,2 9 5 c  ,35 c  .Hencethedesiredparticularsolutionis  24955 5 55 x x yxexe  . 27. Firstwespottheroot1 r  .Thenlongdivisionofthepolynomial3234rr by1 r yieldsthequadraticfactor  22 442rrr ,withroots2,2 r  .Hencethegeneralsolutionis  22 123 x xx y xcececxe  . 28. Firstwespottheroot2 r  .Thenlongdivisionofthepolynomial32252 rrr by thefactor2 r yieldsthequadraticfactor  2 231211 rrrr  ,withroots 1 1, 2 r  .Hencethegeneralsolutionis  22 123 x xx y xcecece  29. Firstwespottheroot3 r  .Thenlongdivisionofthepolynomial327 r  by3 r  yieldsthequadraticfactor239 rr ,withroots333 22 ri  .Hencethegeneralsolutionis  332 123 3333cossin 22 xx y xceecxcx    . 30. Firstwespottheroot1 r  .Thenlongdivisionofthepolynomial43236 rrrr by1 r  yieldsthecubicfactor32236



cos3sin3

xcececxcx  .







rrr

.Nextwespottheroot2 r  ,andanotherlongdivisionyieldsthequadraticfactor23 r  ,withroots3 ri  .Hencethegeneralsolutionis  2 1234

xx y

31. Thecharacteristicequation323480rrr hastheevidentroot1 r  ,andlongdivisionthenyieldsthequadraticfactor  224824rrr

,correspondingtothecomplexconjugateroots22i  .Hencethegeneralsolutionis

2 123cos2sin2 xx y xceecxcx

32. Thecharacteristicequation4323520rrrr hastheroot2 r  ,asisreadily foundbytrialanderror,andlongdivisionthenyieldsthefactorization

3 210rr .Thusweobtainthegeneralsolution   22 1234 x x y xceccxcxe  .

33. Knowingthat3 x y e  isonesolution,wedividethecharacteristicpolynomial 32354rr by3 r andgetthequadraticfactor

2261839rrr .Hencethe generalsolutionis

33 123cos3sin3 xx y xceecxcx 

34. Knowingthat23 x y e  isonesolution,wedividethecharacteristicpolynomial 32 32128 rrr by32 r andgetthequadraticfactor24 r  .Hencethegeneralsolutionis

23 123cos2sin2 x y xcecxcx  .

35. Thefactthatcos2 y x  isonesolutiontellsusthat24 r  isafactorofthecharacteristic polynomial4326525204 rrrr  .Thenlongdivisionyieldsthequadraticfactor

2 6513121 rrrr

23 1234cos2sin2 xx y xcececxcx 

36. Thefactthatsin x y ex  isonesolutiontellsusthat  22 1122rrr  isafactor ofthecharacteristicpolynomial32911414 rrr

factor97 r .Hencethegeneralsolutionis

 2 x y xABxCxDe

186

HOMOGENEOUSEQUATIONSWITHCONSTANTCOEFFICIENTS











 ,withroots11





, 23 r

.Hencethegeneralsolutionis

.Thenlongdivisionyieldsthelinear

 

,sothegeneralsolutionis

 .Impositionofthegiveninitialconditionsyieldstheequations 18,12,213,7 ADBDCDD  withsolution11 A  ,5 B  ,3 C  ,7 D  .Hencethedesiredparticularsolutionis  2 11537 x yxxxe  .

Giventhat5 r  isonecharacteristicroot,wedivide5 r intothecharacteristicpolynomial325100500rrr andgettheremainingfactor2100 r  .Thusthegeneralsolutionis  5cos10sin10 x yxAeBxCx  .Impositionofthegiveninitialconditions yieldstheequations 0,51010,25100250ABACAB  ,

79 123cossin xx y xceecxcx  37. Thecharacteristicequationis  43310 rrrr

38.

43. (a) Givenacomplexnumber zxiy  wedefine r tobe22 x y  and  tobethe uniqueanglesatisfyingcos x

Section3.3 187 Copyright©2015PearsonEducation,Inc. withsolution2 A  ,2 B  ,0 C  .Hencethedesiredparticularsolutionis  5 22cos10 x yxex  . 39. Thecharacteristicpolynomialis  332 26128rrrr  , sothedifferentialequation is61280 yyyy  . 40. Thecharacteristicpolynomialis  232 24248rrrrr  ,sothedifferential equationis2480 yyyy  . 41. Thecharacteristicpolynomialis 2244416rrr , sothedifferentialequationis (4)160yy 42. Thecharacteristicpolynomialis  3 26424124864rrrr

1248640yyyy  

, sothedifferential equationis(6)(4)



y r   ,and  .ThenEuler’sformula gives  cossin i xy reririxiyz rr       .

i e  ;22 i e   ; 332 i ie   ; 124 i ie  

13223 i ie  

r 

,sin

(b) 440

;

  ,thesquarerootsof223 i 

23 i e   44.

24213

ii x

    (b)  2 224324 ,3 22 ii x iii       45.

 3 1212cossincos3sin3 ixix yxcececxixcxix  46. Thecharacteristicpolynomialis 2623 rirriri  ,sothegeneralsolutionis

are

(a)

,2 22

iii

ThecharacteristicpolynomialisthequadraticpolynomialofProblem44(b).Hencethe generalsolutionis

188 HOMOGENEOUSEQUATIONSWITHCONSTANTCOEFFICIENTS Copyright©2015PearsonEducation,Inc.  32 1212cos3sin3cos2sin2 ixix yxcececxixcxix  . 47. Thecharacteristicrootsare 22313rii  ,sothegeneralsolutionis      1313 1212cos3sin3cos3sin3 ixix xx y xcececexixcexix   . 48. Thegeneralsolutionis  x xx y xAeBeCe  ,where13 2 i    and 13 2 i   .Impositionofthegiveninitialconditionsyieldstheequations 22 17 0 0 ABC ABC ABC      thatwesolvefor1 3 ABC .Thusthedesiredparticularsolutionisgivenby    1132132 3 ixix x yxeee    ,which(usingEuler’srelation)reducestothegiven real-valuedsolution. 49. WeadoptthesamestrategyaswasusedinProblem48.Thegeneralsolutionis  2cossin xx yxAeBeCxDx  .Impositionofthegiveninitialconditionsyields theequations 0 20 40 830 ABC ABD ABC ABD       thatwesolvefor2 A  ,5 B  ,3 C  ,and9 D  .Thus  2 253cos9sin xx yxeexx  . 50. If0 x  ,thenthedifferentialequationis0 yy   ,withgeneralsolution cossin y AxBx .Butif0 x  ,thenitis0 yy   ,withgeneralsolution coshsinh yCxDx  .Tosatisfytheinitialconditions  101y  ,  100y   wechoose 1 AC and0 BD .Buttosatisfytheinitialconditions  200 y  ,  201 y   we choose0 AC and1 B D  .Thecorrespondingsolutionsaredefinedby  1 cos,0 cosh,0; xx yx xx       and  2 sin,0 sinh,0 xx yx xx       .

Section3.3 189 Copyright©2015PearsonEducation,Inc. Examinationofleft-andright-handderivativesat0 x  showsnotonlythat  1 yx and  2 yx aredifferentiableat0 x  ,butthat1 y  and2 y  areinfactcontinuousthere. 51. InthesolutionofProblem51inSection3.1weshowedthatthesubstitutionlnvx  gives1dydy y dxxdv   and 22 2222 11 dydydy y dxxdvxdv   .Afurtherdifferentiationusingthechainrulegives 323 333233 231 dydydydy y dxxdvxdvxdv   . Substitutionoftheseexpressionsfor y  , y  ,and y  intothethird-orderEulerequation 320axybxycxydy  ,togetherwithcollectionofcoefficients,yieldsthedesired constant-coefficientequation   32 32320dydydy abacbady dvdvdv  . InProblems52through58welistfirstthetransformedconstant-coefficientequation,thenits

vx  and v ex  . 52. 2 290dy y dv  ; 290 r  ; 3 ri  ;  1212 cos3sin3cos3lnsin3ln yxcvcvcxcx  53. 2 26250dydy y dvdv  ; 26250rr ; 34 ri  ;  33 1212 cos4sin4cos4lnsin4ln v y xecvcvxcxcx      54. 32 3230dydy dvdv  ; 3230rr ; 0,0,3 r  ;  33 123123 ln v yxccvceccxcx  55. 32 32440dydydy dvdvdv  ; 32440rrr ; 0,2,2 r  ;  222 123123ln vv yxccecvecxccx 

characteristicequationandroots,andfinallythecorrespondinggeneralsolutionwithln

SECTION3.4

MECHANICAL VIBRATIONS

Inthissectionwediscussfourtypesoffreemotionofamassonaspring—undamped,underdamped,criticallydamped,andoverdamped.However,theundampedandunderdampedcases—inwhichactualoscillationsoccur—areemphasizedbecausetheyareboththemostinterestingandthemostimportantcasesforapplications.

190 HOMOGENEOUSEQUATIONSWITHCONSTANTCOEFFICIENTS Copyright©2015PearsonEducation,Inc. 56. 3 30dy dv  ; 30 r  ; 0,0,0 r  ;  22 123123lnln y xccvcvccxcx  57. 32 32550dydydy dvdvdv  ; 32440rrr ; 0,33 r  ;     3333333 123123 vv yxccecvecxcxcx   58. 32 32330dydydy y dvdvdv  ; 323310rrr ; 1,1,1 r  ;  212 123123lnln vvv yxcecvecvexccxcx     

1. Frequency:0 1612radsecHz 4 k m    ;period: 0 22 sec 2 P     2. Frequency0 4848radsecHz 0.75 k m    ;period: 0 22 sec 84 P    3. Thespringconstantis1575Nm 0.20m N k  .Thesolutionof3750 xx   with 00 x  and 010 x   is  2sin5 x tt  .Thustheamplitudeis2m,thefrequency is0752.55radsecHz 3 k m    ,andtheperiodis2sec 5  4. (a) With1kg 4 m  and9N0.25m=36Nm k  ,wefindthat012radsec   .Thesolutionof1440 xx   with 01 x  and 05 x   is

5. Thegravitationalaccelerationatdistance R fromthecenteroftheearthis2GM g R  .AccordingtoEquation(6)inthetext,the(circular)frequency  ofa(linearized)pendulum

6. Ifthependulumintheclockexecutes n cyclesperday(86400sec)atParis,thenitsperiodis186400sec p n  .Attheequatoriallocationittakes24hr2min40sec86560sec 

forthesamenumberofcycles,soitsperiodthereis186560sec p n  .Nowlet 13956miR  betheEarth’s“radius”atParis,and2 R its“radius”attheequator.Then substitutionintheequation11 22

p R p R  ofProblem5(with12 LL  )yields23963.33mi R  .

Thusthis(rathersimplistic)calculationgives7.33miasthethicknessoftheEarth’s equatorialbulge.

7. Theperiodequation

3960100.103960100px  yields 1.9795mi10.450ft x  forthealtitudeofthemountain.

8. Let n bethenumberofcyclesrequiredforacorrectclockwithunknownpendulum length1 L andperiod1 p toregister24hrs86400sec  ,so186400 np  .Thegivenclock withlength230in L  andperiod2 p loses10min600sec  perday,so287000 np  .

9. Designating  x t asinthesuggestion,weseethatthemassissubjecttoarestorative force S Fkx  togetherwiththeforceofgravity Wmg  .Wealsoassumethatthe massissubjecttoadampingforce R Fcx   .ApplyingNewton’slawthengives

Section3.4 191

  51312513 cos12sin12cos12sin2cos12 1212131312 xttttt      , where152tan5.8884rad 12  (b) 13

12 C 

0.5236sec 12 T  

1.0833m

and 2

isgivenby

gGM LRL   ,soitsperiodis22 L pR GM     .

2 2



ThentheformulaofProblem5yields111

Lpnp Lpnp 

2 1

L    

222 86400 87000

,so

86400 3029.59in 87000

Thusthebuoyundergoessimpleharmonicmotionaboutanequilibriumof e x h  .Further,withthegivennumericalvaluesof  ,h,andg,theamplitudeofoscillationis

11. ThedifferentialequationfromProblem10mustbemodifiedtoreflectthefactthatthe weightdensityofwateris362.4lbft(asopposedto31gcminthecgssystem).Thus theweightofwaterdisplacedbythebuoyisgivenby2 62.4 rx  .Moreover,themass andweightofthebuoyaregiventobe3.125slugsand100lb,respectively.Applying

192 MECHANICALVIBRATIONS Copyright©2015PearsonEducation,Inc. mxkxmgcx   ,or mxcxkxmg  .Finally,substituting0 y xs ,sothat 0 x ys andthus x y   and x y   ,yields  0 mycykysmg  ,or 0mycykymgks  ,whichisEquation(5)with  F t assumingtheconstantvalue 0mgks . 10. Themassofthebuoyisgivenby2mrh   ,andthenetdownwardforceonthebuoyis 22 Frhgrgx   .(Notethatthedepth  x t istakentobepositive.)Therefore Newton’ssecondlaw maF  gives 222 rhxrhgrgx   , whichsimplifiesto g x xg h   . Thecomplementaryfunctionforthisequationis  1020 cossin c x tctct  ,where 0 g h    .Aparticularsolutionisgivenby  p x tA  ,where A isaconstant,andsub-

  .Thusthegeneralsolutionof

 1020 cossin cp x txtxtctcth  . Applyingtheinitialconditions  000xx   gives1ch  and20 c  .Alltold,the

 cos0 x thth   .

100cm h  andtheperiodis 0 22 22.01sec h p g g h      .

stitutingintothedifferentialequationshowsthat Ah

thedifferentialequationis

motionofthebuoyisgivenby

maF  thengives2 3.12510062.4 x rx   ,or 62.4232 3.125 xrx    .Thefrequency oftheoscillationsofthebuoyistherefore0 2   ,where062.43.125 r    .Sincethefre-

(d) Theorbitalvelocityvofsuchasatellitemustbesuchthatthecentrifugalforce

onthesatellitejustoffsetstheweight mg ofthesatelliteatthesurfaceoftheearth.Thus

c.Thisisnota

t  thesatelliteisdirectlyovertheholeintheearth atthetopofFigure3.4.13,andthatitsorbitproceedsinaclockwisedirection.Wefound inpart c thatthedistance r oftheparticlefromthecenteroftheearthis

Thekeyobservationisthat0 t  istheangledrawnclockwisefromtheverticaltotheradiusvectorofthesatelliteattimet;thus,thedistance rt issimplytheverticalcomponentofthesatellite’sposition.Itfollowsthat rt completesonecyclethroughtheearth (andback)inthesamelengthoftimerequiredforthesatellitetocompleteoneorbit aroundtheearth.

Section3.4 193 Copyright©2015PearsonEducation,Inc. quencyofthebuoy’smotionisobservedtobe4cycles0.4cyclessec 10sec  ,wecanequate thetwotoconcludethat162.40.4 23.125 r    ,whichgives 3.125 0.80.3173ft3.8in 62.4 r   12. (a) Substitutionof 3 r r M M R     in2r r GMm F r  yields3 r GMm Fr R  . (b) Because3GMg R R  ,theequation r mrF   yieldsthedifferentialequation 0 g rr R   (c) Thesolutionofthisequationwith  0 rR  and 00 r   is  cos0 rtRt   ,where 0 g R   .Hence,with2 32.2ftsec g  and39605280ft R  ,wefindthattheperiod oftheparticlessimpleharmonicmotionis 0 225063.10sec84.38min R p g    

2 mv R

2 mv mg R  ,whichimpliesthat 244 32.2ftsec39605280ft2.594710ftsec1.769110mihr vgR . Becausethecircumferenceoftheearthis2 R

,theperiodofthesatellite’sorbitis 2 2 R R g gR    ,whichisequaltotheperiodoftheparticlefoundinpart

coincidence.Imaginethatattime0





cos0 rtRt   .

194 MECHANICALVIBRATIONS Copyright©2015PearsonEducation,Inc. (e) Theparticlepassesthroughthecenteroftheearthwhen  0 cos0rtRt   ,thatis, when02 t    ,or 20 t    .Atthistimethespeedoftheparticleis  4 0000 0 sinsin1.769110mihr 2 g rtRtRRgR R         (f) Inpart d wefoundtheorbitalvelocitytobe vgR  ,inagreementwithpart e Againthisisnotaconcidence.Theverticalcomponentofthesatellite’svelocityvector  t v atanygiventimetisequaltothespeed rt  oftheparticleatthattime.Atthe momentwhentheparticlepassesthroughthecenterofearth,thesatelliteistravelling straightdownward,andhence  t v isvertical.Thereforetheorbitalvelocityvofthe satellite,whichisthemagnitudeof  t v ,isequaltothespeedoftheparticleatthismoment. 13. (a) Thecharacteristicequation  2 109252210 rrrr hasroots 21 , 52 r  Whenweimposetheinitialconditions 00 x  , 05 x   onthegeneralsolution  252 12 tt x tcece  wegettheparticularsolution   50252 xteett  . (b) Thederivative  2252510 25205540 tttt t xeeee   when 5 10ln2.23144 4 t  .Hencethemass’sfarthestdistancetotherightisgivenby 5512 10ln4.096 4125 x     . 14. (a) Thecharacteristicequation  2 22 251022651150 rrr hasroots 1151 3 55 i ri   .Whenweimposetheinitialconditions 020 x  , 041 x   on thegeneralsolution   5cos3sin3 t x teAtBt  weget20 A  ,15 B  .Thecorrespondingparticularsolutionisgivenby    55 20cos315sin325cos3tt xettet t   , where13tan0.6435 4   . (b) Thustheoscillationsare“bounded”bythecurves5 25 t x e  ,andthepseudoperiod ofoscillationis2 3 T   (because3   ).

InProblems15-21thegraphofthedampedmotion  x t ,thatis,withthedashpotattached,is shownasasolidline;thegraphofthecorrespondingundampedmotion

Section3.4 195

ut isdashed.

With damping: Thecharacteristicequation2 1 340 2 rr hasroots2,4 r  .When weimposetheinitialconditions 02 x  , 00 x   onthegeneralsolution  24 12 tt x tcece  wegettheparticularsolution  2442tt x tee  thatdescribes overdampedmotion. Without damping: Thecharacteristicequation2 1 40 2 r  hasroots22 ri  .When weimposetheinitialconditions 02 x  , 00 x   onthegeneralsolution   cos22sin22 utAtBt  wegettheparticularsolution  2cos22 utt  0 1 2 3 −2 0 2 t x Problem 15 0 1 2 −2 0 2 t x Problem 16 16. With damping: Thecharacteristicequation2330630 rr hasroots3,7 r  Whenweimposetheinitialconditions 02 x  , 02 x   onthegeneralsolution  37 12 tt x tcece  wegettheparticularsolution  3742tt x tee  thatdescribes overdampedmotion. Without damping: Thecharacteristicequation23630 r  hasroots21 ri  . Whenweimposetheinitialconditions 02 x  , 02 x   onthegeneralsolution   cos21sin21 utAtBt  wegettheparticularsolution     222 2cos21sin212cos210.2149 2121 utttt 

15.

196 MECHANICALVIBRATIONS Copyright©2015PearsonEducation,Inc. 17. With damping: Thecharacteristicequation28160 rr hasroots4,4 r  .When weimposetheinitialconditions 05 x  , 010 x   onthegeneralsolution  4 12 t x tccte  wegettheparticularsolution  4 521 t xtet thatdescribescriticallydampedmotion. Without damping: Thecharacteristicequation2160 r  hasroots4 ri  .Whenwe imposetheinitialconditions 05 x  , 010 x   onthegeneralsolution  cos4sin4 utAtBt  wegettheparticularsolution   55 5cos4sin45cos45.8195 22 utttt  . 0 1 2 −5 0 5 t x Problem 17 0 1 2 −1 1 t x Problem 18 18. With damping: Thecharacteristicequation2212500 rr hasroots34 ri  . Whenweimposetheinitialconditions 00 x  , 08 x   onthegeneralsolution  3cos4sin4 t x teAtBt  wegettheparticularsolution  3332sin42cos4 2 tt xtetet     thatdescribesunderdampedmotion. Without damping: Thecharacteristicequation22500 r  hasroots5 ri  .When weimposetheinitialconditions 00 x  , 08 x   onthegeneralsolution  cos5sin5 utAtBt  wegettheparticularsolution  883 sin5cos5 552 uttt    

Section3.4 197 Copyright©2015PearsonEducation,Inc. 19. Thecharacteristicequation24201690 rr hasroots56 2 ri  .Whenweimpose theinitialconditions 04 x  , 016 x   onthegeneralsolution   52cos6sin6 t x teAtBt  wegettheparticularsolution   5252 131 4cos6sin6313cos60.8254 33 tt xtettet     thatdescribesunderdampedmotion. Without damping: Thecharacteristicequation241690 r  hasroots13 2 ri  .When weimposetheinitialconditions 04 x  , 016 x   onthegeneralsolution  1313cossin 22 utAtBt   wegettheparticularsolution  133213413 4cossin233cos0.5517 2132132 tt utt  . 0 1 2 −4 4 t x Problem 19 0 1 2 −5 5 t x Problem 20 20. With damping: Thecharacteristicequation2216400 rr hasroots42 ri  Whenweimposetheinitialconditions 05 x  , 04 x   onthegeneralsolution  4cos2sin2 t x teAtBt  wegettheparticularsolution  44 5cos212sin213cos21.1760 tt xtettet  thatdescribesunderdampedmotion.

198 MECHANICALVIBRATIONS

Without damping: Thecharacteristicequation22400 r  hasroots25 ri  .When weimposetheinitialconditions 05 x  , 04 x   onthegeneralsolution   cos25sin25 utAtBt  wegettheparticularsolution     2129 5cos25sin25cos250.1770 55 utttt  21. With damping: Thecharacteristicequation2101250 rr hasroots510 ri  Whenweimposetheinitialconditions 06 x  , 050 x   onthegeneralsolution  5cos10sin10 t x teAtBt  wegettheparticularsolution  55 6cos108sin1010cos100.9273

xtettet  thatdescribesunderdampedmotion.

hasroots55

weimposetheinitialconditions 06 x  , 

  onthegeneralsolution   cos55sin55 utAtBt  wegettheparticularsolution     6cos5525sin55214cos550.6405utttt  0 1 −6 6 t x Problem 21 22. (a) With120.375slug 32 m  ,3lb-secft c  ,and24lbft k  ,thedifferentialequation isequivalentto3241920 xxx  .Thecharacteristicequation23241920 rr hasroots443 ri  .Whenweimposetheinitialconditions 01 x  , 00 x   on

Without damping: Thecharacteristicequation21250 r 

ri  .When

050 x

Section3.4 199 Copyright©2015PearsonEducation,Inc. thegeneralsolution   4cos43sin43 t x teAtBt      wegettheparticularsolution      4 4 4 1 cos43sin43 3 231cos43sin43 322 2 cos43. 36 t t t xtett ett et              (b) Thetime-varyingamplitudeis21.15ft 3  ,thefrequencyis436.93radsec  ,and thephaseangleis6 . 23. (a) With100slug m  weget100 k   .Butwearegiventhat   80cyclesmin21min60sec83    , andequatingthetwovaluesyields7018lbft k  (b) With1782cyclessec 60  ,Equation(21)inthetextyields 372.31lbftsec c  Hence1.8615 2 c p m  .Finally0.01 pt e  gives2.47sec t  30. Intheunderdampedcasewehave  11cossin pt x teAtBt  and  111111 cossinsincosptpt x tpeAtBteAtBt    . Theconditions 00 x x  , 00 x v   yieldtheequations0 Ax  and10 pABv   , whence00 1 vpx B    . 31. Thebinomialseries    23112 11 2!3! xxxx     

33. If 11x xt  and 22x xt  aretwosuccessivelocalmaxima,then12112

200 MECHANICALVIBRATIONS Copyright©2015PearsonEducation,Inc. convergesif1 x  .(See,forinstance,Section10.8ofEdwardsandPenney,Calculus: EarlyTranscendentals,7thedition,Pearson,2008.)With1 2   and 2 4 c x mk  inEq. (22)ofSection3.4inthedifferentialequationstext,thebinomialseriesgives 22242 22 100 222111. 4481288 kckckccc p mmmmkmmkmkmk      32. If  cos1 pt xtCet ,then  111 cossin0 ptpt xtpCetCet   yields  1 1 tan p t    .Becausethetangentfunctionisperiodicwithperiod  and thelocalmaximaandminimaof  x t

adistanceof 1 2  .

tt   ,and so  1 111 cos pt xCet and 22 21211 coscos ptpt xCetCet  .Hence 112 2 ptt x e x  andtherefore  1 12 21 2 ln x p ptt x       . 34. With10.34 t  and21.17 t  wefirstusetheequation12112 tt   fromProblem33 tocalculate127.57radsec 0.83    .Next,with16.73 x  and21.46 x  ,theresultof Problem33yields16.73ln1.84 0.831.46 p     .ThenEquation(16)inthissectiongives 100 221.8411.51lb-secft 32 cmp ,andfinallyEquation(22)yields 222 41189.68lbft 4 mc k m    . 35. Thecharacteristicequation2210 rr hasroots1,1 r  .Whenweimposetheinitialconditions 00 x  , 10 x   onthegeneralsolution  12 t x tccte  wegetthe particularsolution  1 t x tte  . 36. Thecharacteristicequation 2221100 n rr hasroots110 n r  .Whenwe imposetheinitialconditions 00 x  , 10 x   onthegeneralsolution

areinterlaced,successivemaximaareseparatedby

SECTION3.5

NONHOMOGENEOUS EQUATIONS AND UNDETERMINED COEFFICIENTS

Themethodofundeterminedcoefficientsisbasedon“educatedguessing”.Ifwecanguesscorrectlythe form ofaparticularsolutionofanonhomogeneouslinearequationwithconstantcoefficients,thenwecandeterminetheparticularsolutionexplicitlybysubstitutioninthegivendifferentialequation.ItispointedoutattheendofSection3.5thatthissimpleapproachisnotalwayssuccessful—inwhichcasethemethodofvariationofparametersisavailableifacomplementaryfunctionisknown.However,undeterminedcoefficients does turnouttoworkwellwith asurprisinglylargenumberofthenonhomogeneouslineardifferentialequationsthatariseinelementaryscientificapplications.

Section3.4 201 Copyright©2015PearsonEducation,Inc.  12 exp110exp110 nn x tctct     wegettheequations  12120,1101101 nn cccc  withsolution1125nn c  , 1 225nn c  .Thisgivestheparticularsolution    2 exp10exp10 1010sinh10 2 nn ntntn tt x teet      37. Thecharacteristicequation 2221100 n rr hasroots110 n ri  .Whenwe imposetheinitialconditions 00 x  , 10 x   onthegeneralsolution  cos10sin10 tnn x teAtBt      wegettheequations10 c  ,12101 n cc  withsolution10 c  ,210 n c  .Thisgives theparticularsolution  310sin10ntn x tet  . 38. Thisfollowsfrom    2 sinh10 limlim10sinh10lim 10 n ntntt n nnn t x tettete t   and    3 sin10 limlim10sin10lim 10 n ntntt n nnn t x tettete t   , usingthefactthat 00 sinsinh limlim0     (byL’Hôpital’srule,forinstance).

IneachofProblems1-20wegivefirsttheformofthetrialsolutiontrial y ,thentheequationsin thecoefficientswegetwhenwesubstitutetrial y intothedifferentialequationandcollectlike terms,andfinallytheresultingparticularsolution

202 NONHOMOGENEOUSEQUATIONSANDUNDETERMINEDCOEFFICIENTS

p y 1. 3 trial x yAe  ;251 A  ; 13 25 x p ye  . 2. trial y ABx ;24 AB  ,23 B  ;  1 56 4 p yx  . 3. trialcos3sin3 y AxBx ;1530 AB  ,3152 AB ; 15cos3sin3 3939 p y xx . 4. trial x x yAeBxe  ;9120 AB ,93 B  , 41 93 x x p yexe  5. Firstwesubstitute1cos2 2 x for2 sin x ontheright-handsideofthedifferentialequation,leadingtotrialcos2sin2 y ABxCx ,andthen 11 ,32,230 22 ABCBC  ; 131cos2sin2 22613 p yxx  6. 2 trial yABxCx  ;7440 ABC ,780 BC ,71 C  ; 4812 343497 p yxx  7. Firstwesubstitute 2 x x ee forsinh x ontheright-handsideofthedifferentialequatio, leadingtotrial x x y AeBe ; 1 3 2 A  , 1 3 2 B  ; 111sinh 663 xx p y eex .(Note thataccordingtoRule1inthetext,wecouldalsohavestartedwith trialcoshsinh y AxBx .) 8. Firstwenotethat 22 cosh2 2 x x ee x   ispartofthecomplementaryfunction 22 c12 x x ycece  ,leadingto  22 trial x x yxAeBe  ; 1 8 A  , 1 8 B  ;

Firstwenotethat

ispartofthecomplementaryfunction3 12 x x c y cece .Then  trial x yAxBCxe  ,andthen 3142081 ABCC  ; 111 3168 x p y xxe   .

Firstwenotetheduplicationwiththecomplementaryfunction12cos3sin3 c y cxcx . Then trialcos3sin3 yxAxBx  ;62 B  ;63 A  ;  111 cos3sin32sin33cos3 326 p y xxxxxxx     11. Firstwenotetheduplicationwiththecomplementaryfunction 123cos2sin2 c y ccxcx .Then  trial yxABx  ;41 A  ,83 B  ; 131232 488 p y xxxx     . 12. Firstwenotetheduplicationwiththecomplementaryfunction123cossin c y ccxcx . Then trialcossin yAxxBxCx  ;2 A  ,20 B  ,21 C  ; 1 2sin 2 p y xxx 13. trialcossin x yeAxBx  ;740 AB ,471 AB  ;  1 7sin4cos 65 x p yexx  14. Firstwenotetheduplicationwiththecomplementaryfunction  1234 x x c yccxeccxe  .Then  2 trial x yxABxe  ;8240 AB ,241 B  ; 223 111 3 82424 x xx p y xxexexe     15. Thisissomethingofatrickproblem.Wecannotsolvethecharacteristicequation 54510rr tofindthecomplementaryfunction,butwecanseethatthecomplementaryfunctioncontainsnoconstantterm(why?).Hencewecantaketrial y A  ,leading immediatelytotheparticularsolution17 p y 

x e

10.

18. Firstwenotetheduplicationwiththecomplementaryfunction 22

19. Firstwenotetheduplicationwiththepart12ccx  ofthecomplementaryfunction(which correspondstothefactor2 r ofthecharacteristicpolynomial).Then

20. Firstwenotethatthecharacteristicpolynomial3rr hasthezero1 r  correspondingto theduplicatingpart x e ofthecomplementaryfunction.Then:trial x y AxBe ;

InProblems21-30welistfirstthecomplementaryfunction yc ,thentheinitiallyproposedtrial functioni y ,andfinallytheactualtrialfunction p y ,inwhichduplicationwiththecomplementaryfunctionhasbeeneliminated.

21. c12cossin x yecxcx  ;

; 95,18620,18120,182 ABCDCDD  ;  23332351211 4569 98127981 x xxx p y xxeexexe    

Firstwenotetheduplicationwiththecomplementaryfunction12cossin c y cxcx . Then  trialcossin yxABxxCDxx   ; 2204122140 BCDADB  ; 1112 cossinsincos 444 p y xxxxxxxx    

trial x y ABCxDxe

17.

1234

 .Then  2 trial x x yxAexBCxe  ; 61,12380,241 ABCC  ;  222211911 24196 614424144 x xxxx p y xexxexexexe    

x xxx c ycececece

 22 trial yxABxCx  ; 4121,12480,243 ABBCC  ;  22234 5111 104 4288 p y xxxxxx     .

7 A  ;31 B  ; 1 7 3 x p y xe 

icossin

yeAxBx  ;  cossin x p yxeAxBx 

22.  2 12345 x x c y ccxcxcece ;  2 i x y ABxCxDe ;  32 x p yxABxCxxDe 

23. 12 cos2sin2 c y cxcx ;  icos2sin2 yABxxCDxx  ; cos2sin2 p yxABxxCDxx  

24. 34 123 x x c yccece  ;  3 i x yABxCDxe  ;  3 x p yxABxxCDxe 

25. 2 12 x x c ycece  ;

2 i x x yABxeCDxe  ;

2 x x p yxABxexCDxe 

26.

3 12 cos2sin2 x c yecxcx  ;

33 icos2sin2 xx yABxexCDxex  ;

33 cos2sin2 xx p yxABxexCDxex

27.  1234 cossincos2sin2 c ycxcxcxcx  ;

icossincos2sin2 yAxBxCxDx  ;

cossincos2sin2 p y xAxBxCxDx 

28.  1234cos3sin3 c yccxcxcx  ;

22 icos3sin3y ABxCxxDExFxx  ;

22 cos3sin3 p y xABxCxxDExFxx 

29. 222 12345 x xx c yccxcxecece

InProblems31-40welistfirstthecomplementaryfunction yc ,thetrialsolutiontr y forthe methodofundeterminedcoefficients,andthecorrespondinggeneralsolutiong

, where

Section3.5 205







 









 ;  22 i x yABxeCeDexx  ;  322 p x yxABxexCexDexx  30.  1234 x x c yccxeccxe  ;  22 icossin p y yABxCxxDExFxx 

yyycp 

yx .

y resultsfromdeterminingthecoefficientsintr y soastosatisfythegivennonhomogeneousdifferentialequation.Thenwelistthelinearequationsobtainedbyimposingthegiven initialconditions,andfinallytheresultingparticularsolution

206 NONHOMOGENEOUSEQUATIONSANDUNDETERMINEDCOEFFICIENTS Copyright©2015PearsonEducation,Inc. 31. 12 cos2sin2 c y cxcx ;tr y ABx ;g12cos2sin2 2 x ycxcx  ;11 c  ,2 1 22 2 c  ; ()cos2(3/4)sin2/2 yxxxx  32. 2 12 x x c ycece  ;tr x yAe  ; 2 g12 1 6 x xx ycecee  ;12 1 0 6 cc , 12 1 23 6 cc  ;  2 581 236 x xx y xeee  33. 12 cos3sin3 c y cxcx ;trcos2sin2 y AxBx ;g12 1 cos3sin3sin2 5 ycxcxx  ; 11c  ,2 2 30 5 c  ,  21 cos3sin3sin2 155 y xxxx  34. 12cossin c y cxcx ; trcossin yxAxBx  ;g12 1 cossinsin 2 y cxcxxx  ; 11c  ,21 c  ,  1 cossinsin 2 yxxxxx  35.  12cossin x c yecxcx  ;tr y ABx ; g12cossin1 2 x x yecxcx ;113 c  , 12 1 0 2 cc ;  5 2cossin1 22 x x yxexx     36. 22 1234 x x c y ccxcece ;  22 tr yxABxCx  24 22 g12341648 xx x x yccxcece 1342343434 1 1,221,441,881 8 cccccccccc   222439531111 32464641648 xx y xxeexx  37. 123 x x c yccecxe  ;  2 tr x yxAxBCxe  23 g123 11 26 x xxx y ccecxexxexe  1223230,10,211cccccc   2311 443 26 x y xxexxx  

Section3.5 207 Copyright©2015PearsonEducation,Inc. 38.  12cossin x c yecxcx  ;trcos3sin3 y AxBx g12 67 cossincos3sin3 8585 x yecxcxxx  112 621 2,0 18585 ccc  17619767cossincos3sin3 85858585 x y xexxxx     39. 123 x c y ccxce ;   2 tr x yxABxxCe  23 g12326 x x xx y ccxcexe  132331,10,31ccccc     123181834 6 x y xxxxxe  40. 1234tr cossin; xx c ycececxcxyA  g1234cossin5 xx ycececxcx  123124123124 50,0,0,0 cccccccccccc    1 5510cos20 4 xx yxeex  41. Thetrialsolution2345 tr yABxCxDxExFx  leadstotheequations 226240 226241200 2312600 24200 250 28 ABCDE BCDEF CDEF DEF EF F       thatarereadilysolvedbyback-substitution.Theresultingparticularsolutionis  23452554503020104 yxxxxxx  . 42. Thecharacteristicequation43220 rrrr hasroots1,2 r  ,and i  ,sothe complementaryfunctionis2c1234cossin xx y cececxcx  .Wefindthatthecoefficientssatisfytheequations

43. (a) ApplyingEuler’sformulagives  33223 cos3sin3cossincos3cossin3cossinsin x ixxixxixxxxix

323 cos3cos1cos4cos3cos x xxxx

44 coscoscos3 x xx .Theformulafor3 sin x isderived similarlybyequatingimaginarypartsinthefirstequationabove.

(b) Uponsubstitutingthetrialsolutioncossincos3sin3 p yAxBxCxDx

208 NONHOMOGENEOUSEQUATIONSANDUNDETERMINEDCOEFFICIENTS Copyright©2015PearsonEducation,Inc. 123 124 123 124 2550 4500 4600 81200 ccc ccc ccc ccc     . Solutionofthissystemgivesfinallytheparticularsolutioncp

 ,wherep y

particularsolutionofProblem41and 2 c 1035210cos390sin xx yeexx  .

 . Whenweequaterealpartswegettheequation  



y yy

isthe

andreadilysolvefor331

inthe

y yxx   ,wefindthat 11 ,0,,0 420 ABCD  . Theresultinggeneralsolutionis  12 11 cos2sin2coscos3 420 yxcxcxxx  . 44. Weusetheidentity11 22 sinsin3cos2cos4 x xxx  ,andhencesubstitutethetrialsolutioncos2sin2cos4sin4 p y AxBxCxDx  inthedifferentialequation 11 22cos2cos4 y yyxx .Wefindthat 31152 ,,, 2613482241 ABCD  . Theresultinggeneralsolutionis    2 12 3311 cossin3cos22sin215cos44sin4 2226482 x y xecxcxxxxx     . 45. Wesubstitute    2 42111 sin1cos212cos2cos234cos2cos4 448 x xxxxx



differentialequation31 44 4coscos3

eindependentsolutions1 y and2 y oftheassociatedhomogeneous equation,theirWronskian

Section3.5 209 Copyright©2015PearsonEducation,Inc. ontheright-handsideofthedifferentialequation,andthensubstitutethetrialsolution cos2sin2cos4sin4 p y AxBxCxDxE  . Wefindthat 111 ,0,,0, 105624 ABCDE  Theresultinggeneralsolutionis 12 111 cos3sin3cos2cos4 241056 ycxcxxx  46. Bytheformulafor3 cos x inProblem43,thedifferentialequationcanbewrittenas 31 coscos3 44 yyxxxx   . Thecomplementarysolutionis12cossin c y cxcx ,sowesubstitutethetrialsolution  cossincos3sin3 p yxABxxCDxxEFxxGHxx    Wefindthat 3313 ,0,,0,,,0 161632128 ABCDEFGH  . Hencethegeneralsolutionisgivenby12 c y yyy ,where  2 1 1 3cos3sin 16 yxxxx  and  2 1 3sin34cos3 128 yxxx  InProblems47–49welistth



WWyy  ,thecoefficientfunctions    2 1 yxfx uxdx Wx     and    1 2 yxfx uxdx Wx     intheparticularsolution1122 p y uyuy ofEq.(32)inthetext,andfinally p y itself. 47. 2 1 x ye  ,2 x ye  , 3 x We  , 3 1 4 3 x ue  , 2 22 x ue  , 2 3 x p ye  48. 2 1 x ye  , 4 2 x ye  , 62 x We  ,12 x u  , 6 2 1 12 x ue  ,  12 61 12 x p yxe  49. 2 1 x y e  , 2 2 x y xe  , 4 x We  , 2 1 ux  ,22ux  , 22 x p yxe 

12 ,