《高數(shù)雙語》課件section 6.4

Section7.4Higher-OrderLinearDifferentialEquations1Anth-OrderLinearDifferentialEquation2Generally,foranth–orderdifferentialequation,iftheunknownfunctionandeachofitsderivativesappearlinearly,thentheequationiscalledanth–orderlineardifferentialequation(n階線性微分方程),ornth–orderlinearequation.LinearDifferentialEquation(LDE)Anth

–orderlineardifferentialequationisandeachofitsderivativeswheretheunknownfunctionappearlinearly.Anth-OrderLDE3thentheequationiscalledanth

–orderhomogeneous

Iflineardifferentialequation(n階齊次線性微分方程);Ifisnotidenticallyzero,thentheequationiscalledanth–ordernonhomogeneouslineardifferentialequation(n階非齊次線性微分方程);Anth

–orderlineardifferentialequationisandeachofitsderivativeswheretheunknownfunctionappearlinearly.Anth-OrderLDEThegeneralformofthesecond–orderlineardifferentialequationisanditsfirstandsecond–orderwheretheunknownfunctionderivativesallenteraslinearterms.lineardifferentialequation(二階齊次線性微分方程);thentheequationiscalledsecond–orderhomogeneous

IfIfisnotidenticallyzero,thentheequationiscalledasecond–ordernonhomogeneouslineardifferentialequation(二階非齊次線性微分方程).4Anth-OrderLDEForexampleisnotalineardifferentialequation.wherep,qareconstant

isasecond–orderhomogeneouslinearequation.isasecond–ordernonhomogeneouslinearequation.Wehadlearnedhowtofindthesolutionofsometypeoffirstdifferentialequationsismuchharderthanbefore.order

differentialequations.Buttofindthesolutionofhighorder5SuperpositionofSolutionsofLDEForconvenienceinwriting,weintroducethesymbolsothanthenth–orderhomogeneouslineardifferentialequation(LDE)maybewrittenmoresuccinctlyasandLinearDifferentialOperator6Itiseasytosee,thatLhasthefollowingproperties:3.1.,ifcisanarbitraryconstant;2.arearbitraryconstants.whereTheorem(PrincipleofSuperpositionofSolutions)aresolutionsofahomogeneouslinearequationthenIfareallconstants.isalsoitssolution,whereSuperpositionofSolutionsofLDE7SuperpositionofSolutionsofLDE8Theorem(PrincipleofSuperpositionofSolutions*)aretwosolutionsofasecond-orderhomogeneousLDEIfarebothconstants.isalsoitssolution,wherethenanylinearcombinationofbothsolutions,ExampleLinearDependenceandIndependenceofFunctions9Definition

SupposethatfunctionsaredefinedonanintervalI.whichareIfthereexistnconstantsnotallzerosuchthatissaidtobe

linearlyindependent(線性無關(guān))holdsforallthenthesetoffunctionsholdsonlyifallissaidtobelinearlydependent(線性相關(guān))

onI;thenthesetoffunctionsonI.IfthisequationTheorem(Conditionsforlinearindependentofsolution)besolutionofanth–orderhomogeneousLDE,LetwhicharedefinedontheintegralI.Thenecessaryand

sufficientconditionforthemtobelinearlyindependentonIisthatonlywhenallthescalarsciarezero.10LinearDependenceandIndependenceofFunctionsLinearDependenceandIndependenceofFunctions11ExampleProvethattwofunctionsandarelinearlyindependentonI=(?∞,+∞).SolutionAssumethatthereexisttwoconstantsc1andc2suchthatonI=(?∞,+∞)TakingandwehaveThenTwofunctionsandarelinearlyindependentonI.12LinearDependenceandIndependenceofFunctionsFortwocontinuousfunctionsf1(x)andf2(x)onanintervalI,assumef1(x)0onI,theyarelinearlydependentifandonly

ifthereexistsaconstantcsuchthatlinearlydependentLinearDependenceandIndependenceofFunctionsTheorem(Testforlinearindependentofsolution)besolutionofthenth–orderhomogeneouslinearLetequation,whicharedefinedontheintervalI.TheWronskiandeterminantconsistingofthevariousderivativesofthesesolutionsisIftheWronskianisnotidenticallyzero,thesolutionsarelinearindependent.IfitisidenticallyzeroovertheintervalI,thesolutionsarelinearlydependentontheinterval.13LinearDependenceandIndependenceofFunctionsExampleProvethattwofunctionscos

2tandsin2tarelinearlyindependentonI=(?∞,+∞).14SolutionHence,functionscos

2tandsin2tarelinearlyindependentonI=(?∞,+∞).StructureoftheGeneralSolutionofaHomogeneousLDEarenlinearindependentparticularTheorem

Ifsolutionsofanth–orderhomogenouslinearequation,theneverysolutionofthelinearequation,y,canbeexpressedasarearbitraryconstants.where15StructureoftheGeneralSolutionofaHomogeneousLDE16Theorem

Letbeanyparticularsolutionofthen-thnonhomogeneouslinearequation(NLDE),thenthesolutionoftheequationcanbeexpressedbyStructureoftheGeneralSolutionofaHomogeneousLDETheorem

Ifaresolutionsofthenonhomogeneouslinearandequationsandmustbethesolutionoftheequationthen17Example18ExampleSupposeisaparticularsolutionofasecond-orderhomogeneouslinearequation,andaretwosolutionsofitscorrespondinghomogeneouslinearequation.Findthegeneralsolutionofthissecond-ordernonhomogeneouslinearequation.SolutionSection7.5Higher-OrderLinearEquationswithConstantCoefficientsSolutionofHigher–OrderHomogeneousLDEwithConstantCoefficientswhereareallconstants.Thegeneralformofthesecond–orderhomogeneousLDEwithconstantcoefficientsisiscalledalineardifferentialTheequationequationwithconstantcoefficientsorlinearequationwithconstantcoefficients.whereareallconstants,whereλ

canberealandcomplex.Tosolvethisequation,wetrytofindasolutionas20SolutionofHigher–OrderHomogeneousLDEwithConstantCoefficientstheremusthaveThesolutionofcharacteristicequationiscalledeigenvaluesorCharacteristicequation(特征方程)Sincecharacteristicroots(特征根).21intothedifferentialequation,wehaveIfwesubstituteHenceSolutionofsecondr–OrderHomogeneousLDEwithConstantCoefficientsObviously,foranycharacteristicroot,theremusthaveasolutionWewillshowthesolutionofthesecond–orderLDEwithrespecttothecharacteristicrootsinthreecases.22CharacteristicequationTheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseIIftherehavetwodifferentrealroots,λ

1andλ2,ofthecharacteristicequation,thenwehaveand23aretwosolutionsoftheequation(1),andsoarelinearlyindependent.TheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseITherefore,allthesolutionofthesecond–orderLDE(1)canbeexpressedasareallconstants.whereand24TheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseIExampleFindthesolutionoftheequationSolutionThecharacteristicequationofthisequationisThen,wehaveandTherefore,thesolutioncanbeexpressedasFinish.25TheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseII26wecanonlyfindoneparticularsolutionIftherehavetworepeatedrealroots,Howtofindanothersolutiony2hatislinearlyindependentofy1?TheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseII27Substitutingy2intotheequation(1),wehaveItiseasytogetthegeneralsolutionoftheaboveequationisTheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseII28TheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseIIExampleFindthesolutionsoftheequationSolutionthenthesolutionThecharacteristicequationofthisequationisofthisequationisFinish.29WeobtaintherootsareTheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseIII3)IfthecharacteristicequationhasapairofconjugatecomplexrootsThen,weknowthatthehomogeneousequationhastwoparticularsolutionsandand30TheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseIIIWecangettherealandimaginarypartsofy1andy2byByEulerformulae,wecanrewritethistwosolutionsasandand31TheSolutionofSecond–OrderLDEwithConstantCoefficients:CaseIIIThen,wecanexpressallthesolutionas32oftheequation(1),andtheyarelinearlyindependent.ThereforeandarealsothesolutionsTheSolutionofSecond–OrderDifferentialEquationwithConstantCoefficients:CaseIIIExampleFindthesolutionoftheequationSolutionThecharacteristicequationofthisequationisTherefore,thesolutionisThen,Example

FindthesolutionoftheequationSolutionThecharacteristicequationofthisequationisTherefore,thesolutionisThen,Finish.Finish.33SolutionofHigher–OrderHomogeneousLDEwithConstantCoefficientsTotheequationwehaveitscharacteristicequation:thesolutionmusthavethetermisasinglerootofthecharacteristicequation,thesolution1)Ifisamultiplecomplexrootoforderthen3)Ifmusthavethetermmusthavethetermthenthesolution2)Ifisamultiplerootoforder34ThenSolutionofHigher–OrderHomogeneousLDEwithConstantCoefficientsExampleFindthesolutionoftheequationSolutionThecharacteristicequationofthisequationisTherefore,thesolutionisThen,ExampleFindthesolutionoftheequationSolutionThecharacteristicequationofthisequationisTherefore,thesolutionisThen,Finish.Finish.35SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsTheorem

Letbeanyparticularsolutionofannth-ordernonhomogeneouslineardifferentialequationbethegeneralsolutionoftheandThen,thegeneralsolutionofthecorrespondinghomogeneousequation.nonhomogeneousequationisThistheoremsaysthattofindthesolutionofnonhomogeneouslinear(2)Findaparticularsolutionofnonhomogeneouslinearequation.(1)Findthehomogeneoussolutionsofhomogeneouslinearequation;Thenwecombinethemtogethertoformthesolution.differentialequationswithconstantcoefficientscanfollowtwosteps:36SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsSolutionThecorrespondinghomogeneousequationisExample

FindthesolutionofandtheeigenvalueareThecharacteristicequationisthenthegeneralsolutionofthehomogeneousequationisisasolutionoftheMoreover,itiseasytoseethatTherefore,thegeneralsolutionisnonhomogeneousequation.Finish.37SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsLetusbeginfromthesecond–ordernonhomogeneousLDE,whereμisaconstantand(1)If38or(2)If,whereμandν

areconstantsandwhereSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsWecanexpecttheparticularsolutionoftheequationasisapolynomialisdeterminedbytheequation.where,whereμisaconstantand(1)If39thenisaparticularsolutionoftheequation,Ifassume

SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients,whereμisaconstantand(1)If40Then,Substitutingthemintotheequation,wehavemthpolynomial?SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsSinceBycomparingthecoefficients,wecandetermine(a)Ifμ

isnotaneigenvalueofthecharacteristicequation.canbeassumedas41SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients42comparethecoefficientsSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients(b)Ifμ

isasingleeigenvalueofthecharacteristicequation.WehaveisThenwecanassumethat43Bycomparingthecoefficients,wecandetermine(c)Ifμ

isarepeatedeigenvalueofthecharacteristicequation.WehaveisThenwecanassumethatSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients44Bycomparingthecoefficients,wecandetermine,whereμisaconstantand(1)IfSumming-uptheparticularsolutioncanbeassumeaswhereSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients45SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsSolutionExampleFindthegeneralsolutionoftheequationThecharacteristicequationofthecorrespondinghomogenousequationisTheeigenvaluesofthecharacteristicequationareThenthegeneralsolutionofthehomogeneousequationisSince

μ

=2isasimpleeigenvalue,weassumethattheparticularsolutionforthenonhomogeneousequationis

46SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsSolution(continued)47ExampleFindthegeneralsolutionoftheequationWesubstitutethemintothenonhomogeneousequation,thenItiseasytogetSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsSolution(continued)ComparingcoefficientsofthepowerswiththesamedegreeonbothHence,thedesiredparticularsolutionissideswehaveandthegeneralsolutionofthegivenequationisFinish.48ExampleFindthegeneralsolutionoftheequationSolutionThecorrespondinghomogeneousequationis,therefore,thegeneralThisequationhasarepeatedeigenvaluesolutionofhomogeneousequationisweassumethatthethenwesubstituteparticularsolutionisitintothenonhomogeneousequationExampleFindthesolutionsofSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsisSincenotaneigenvalueofthecharacteristicequation,49Byrearrangingtheterms,wehavethenThisgivesthatTherefore,theparticularsolutionofnonhomogeneousequationisandthegeneralsolutionisSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsFinish.50SolutionThecharacteristicequationofthecorrespondinghomogeneousthenithasarepeatedeigenvalueExample

Findthesolutionsofwhereaisarealconstant.equationisTherefore,thegeneralsolutionofthehomogeneousisSupposethattheparticularsolutionofthenonhomogeneousequationisSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients51SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsTherefore,thegeneralsolutionofthenonhomogeneousequationisSolution(continued)Bysubstituting,weobtainthatFinish.52Notes53FindthesolutionofwhereNotes54Findthesolutionofwhere4)Bycomparingthecoefficients,determine5)ThesolutionofisSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsInthiscase,theequationcanbeseenastherealorimaginarypart

ofdifferentialequationwithacomplexconstantasWecanexpectthattheparticulararerealconstants.whereandsolutionshouldbeincomplexformasor(2)If,whereμandν

areconstantsandwhere55SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientswillbethesolutionofItiseasytoseethatandandrespectively.Bythisway,theproblemshasbeenchangedintofindthesolutionandthisistheproblemwehavediscussedin(1),excepttheconstantofnonhomogeneouslineardifferentialequationofμ±iν

hereareacomplexconstant.Thenwecanusethesamewayin(1)tofindthesolutionin(2).56SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsExampleFindthegeneralsolutionoftheequationSolution57SincewehaveandNow,wetrytosolvetheequationSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients58ThecharacteristicequationofequationisThen,Therefore,thegeneralsolutionofisSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients59Sincethecomplexnumberisnotaneigenvalueofthecharacteristicequation,thatis,wecanassumethatSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients604)Bycomparingthecoefficients,determineSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients615)ThesolutionofisSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients62Therefore,thegeneralsolutionofthenonhomogeneousequationisFinish.ExampleFindthegeneralsolutionoftheequationSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsTheMethodofUndeterminedCoefficientsWewanttofindaparticularsolutionoftheequationWemayassumethesolutionisherek=0ifμ±iν

arenoteigenvalues;k=1ifμ±iν

areeigenvalues.

Thenwesubstituteitintothenonhomogeneousequation,bythemethodofcomparingcoefficientswecandeterminethecomplexpolynomialZ(x)whosedegreeisthesameasthedegreeof63ImaginarySolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsThus,substitutingZ(x)=R(x)+iI(x)intotheequationofy*:wehave64Z(x)isacomplexpolynomial.AssumethatZ(x)=R(x)+iI(x),whereR(x)andI(x)botharerealpolynomials.RealSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsWefindandarerespectivelytheparticularsolutionsoftheequationsThus,wecanassumetheparticularsolutionofaboveequationis65andrealSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsExampleFindtheparticularsolutionoftheequationwhichsatisfiestheinitialconditions66SolutionThecharacteristicequationofequationisThen,Therefore,thegeneralsolutionisSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients67AssumethatSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients68SubstitutingthemintotheequationwehaveThenSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients69Therefore,wegetaparticularsolutionasThegeneralsolutionoftheequationisSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficients705)Findtheparticularsolutionwiththeinitialconditions.SincetheinitialconditionsarewehaveHenceTheparticularsolution,whichsatisfiestheinitialconditions,isFinish.SolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsExampleFindaparticularsolutionoftheequation71SolutionSolutionofHigher–OrderNonhomogeneousLDEwithConstantCoefficientsExampleFindaparticularsolutionoftheequation72SolutionSection7.6*Euler’sDifferentialEquation73GeneralSolutionofHigher–OrderNonhomogeneousLinearDifferentialEquationswithVariableCoefficientsTheproblemtosolveahigher–ordernonhomogeneouslineardifferentialequationswithvariablecoefficientsismuchdifficultthanthatwithconstantscoefficients.Euler’sdifferentialequationThegeneralformoftheEuler’sdifferentialequationisareallconstants.whereTosolvethisequation,letorthenEuler’sequationcanbechangedintodifferentialequationwithconstantscoefficients.74GeneralSolutionofHigher–OrderNonhomogeneousLinearDifferentialEquationswithVariableCoefficientsExampleFindthegeneralsolutionoftheequationSolutionThisisanEulerdifferentialequation.Letorsothat75GeneralSolutionofHigher–OrderNonhomogeneousLinearDifferentialEquationswithVariableCoefficientsExampleFindthesolutionofSolutionThisequationisanEuler’sequation,thenweletorThenSubstitutethesebacktotheoriginalequation,wehave76GeneralSolutionofHigher–OrderNonhomogeneousLinearDifferentialEquationswithVariableCoefficientsSolution(continued)ThecorrespondinghomogeneousequationisItscharacteristicequationisanditseigenvalueareTherefore,thegeneralsolutionofthehomogeneousequationis77GeneralSolutionofHigher–OrderNonhomogeneousLinearDifferentialEquationswithVariableCoefficientsSolution(continued)AssumethattheparticularsolutionofthenonhomogeneousequationiswehaveandSubstitutethembackintothedifferentialequationwithconstantsTherefore,wehavecoefficients,wehavethen78ReviewStructureofSolutionsofLinearDifferentialEquationsSolutionofHigher–OrderHomogeneousLinearDifferentialEquationswithConstantCoefficientsSolutionofHigher–OrderNonhomogeneousLinearDifferentialEquationswithConstantCoefficients*SolutionofEuler’sDifferentialEquation79Section7.7ApplicationsofDifferentialEquations80MathematicalModelsMathematicalModelisanidealizationofthereal-worldphenomenonandneveracompletelyaccuraterepresentation.81Real-worlddataModelPredications/explanationsMathematicalconclusionsSimplificationAnalysisInterpretationVerificationSomeApplicationsforDifferentialEquations82Ingeneral,theproceduresforapplyingdifferentialequationstosolvepracticalproblemsarethefollowing:(1)Establishtheapproximatedifferentialequationandinitialconditionsusingknowledgeofmathematicsandrelatedsciences;(2)Findthegeneralsolutionoftheequationandthendeterminethedesiredparticularsolutionusingtheinitialconditions.ExampleFindtheequationofthecurvesuchthatthedistancebetweenany