麻省理工開放課程：微分方程 Differential Equations 英文版 DVD 只於電腦播放

課程介紹：
微分方程是一門表述自然法則的語言。理解微分方程解的性質，是許多當代科學和工程的基礎。常微分方程（ODE's）是關於單變量的函數，一般可以認為是時域變量。學習內容包括：利用解釋、圖形和數值方法求解一階常微分方程，線性常微分方程，尤指二階常係數方程，不定係數和參變數，正弦和指數信號：振動、阻尼和共振，複數和冪，傅立葉級數，週期解，Delta函數、卷積和拉普拉斯變換方法，矩陣和一階線性系統：特徵值和特徵向量，非線性獨立系統：臨界點分析和相平面圖。

導師介紹

ArthurMattuckisatenuredProfessorofMathematicsattheMassachusettsInstituteofTechnology.Hemaybebestknownforhis1998book,IntroductiontoAnalysis(ISBN013-0-81-1327)andhisdifferentialequationsvideolecturesfeaturedonMIT'sOpenCourseWare.Insidethedepartment,heiswellknowntograduatestudentsandinstructors,ashewatchesthevideotapesofnewrecitationteachers(anMIT-wideprograminwhichthedepartmentparticipates).
目錄:


Lecture01:Thegeometricalviewofy'=f(x,y):directionfields,integralcurves.
Lecture02:Euler'snumericalmethodfory'=f(x,y)anditsgeneralizations.
Lecture03:Solvingfirst-orderlinearODE's;steady-stateandtransientsolutions.
Lecture04:First-ordersubstitutionmethods:BernouilliandhomogeneousODE's.
Lecture05:First-orderautonomousODE's:qualitativemethods,applications.
Lecture06:Complexnumbersandcomplexexponentials.
Lecture07:First-orderlinearwithconstantcoefficients:behaviorofsolutions,useofcomplexmethods.
Lecture08:Continuation;applicationstotemperature,mixing,RC-circuit,decay,andgrowthmodels.
Lecture09:Solvingsecond-orderlinearODE'swithconstantcoefficients:thethreecases.
Lecture10:Continuation:complexcharacteristicroots;undampedanddampedoscillations.
Lecture11:Theoryofgeneralsecond-orderlinearhomogeneousODE's:superposition,uniqueness,Wronskians.
Lecture12:Continuation:generaltheoryforinhomogeneousODE's.Stabilitycriteriafortheconstant-coefficientODE's.
Lecture13:FindingparticularsolutionstoinhomogeneousODE's:operatorandsolutionformulasinvolvingexponentials.
Lecture14:Interpretationoftheexceptionalcase:resonance.
Lecture15:IntroductiontoFourierseries;basicformulasforperiod2(pi).
Lecture16:Continuation:moregeneralperiods;evenandoddfunctions;periodicextension.
Lecture17:FindingparticularsolutionsviaFourierseries;resonantterms;hearingmusicalsounds.
Lecture19:IntroductiontotheLaplacetransform;basicformulas.
Lecture20:Derivativeformulas;usingtheLaplacetransformtosolvelinearODE's.
Lecture21:Convolutionformula:proof,connectionwithLaplacetransform,applicationtophysicalproblems.
Lecture22:UsingLaplacetransformtosolveODE'swithdiscontinuousinputs.
Lecture23:Usewithimpulseinputs;Diracdeltafunction,weightandtransferfunctions.
Lecture24:Introductiontofirst-ordersystemsofODE's;solutionbyelimination,geometricinterpretationofasystem.
Lecture25:Homogeneouslinearsystemswithconstantcoefficients:solutionviamatrixeigenvalues(realanddistinctcase).
Lecture26:Continuation:repeatedrealeigenvalues,complexeigenvalues.
Lecture27:Sketchingsolutionsof2x2homogeneouslinearsystemwithconstantcoefficients.
Lecture28:Matrixmethodsforinhomogeneoussystems:theory,fundamentalmatrix,variationofparameters.
Lecture29:Matrixexponentials;applicationtosolvingsystems.
Lecture30:Decouplinglinearsystemswithconstantcoefficients.
Lecture31:Non-linearautonomoussystems:findingthecriticalpointsandsketchingtrajectories;thenon-linearpendulum.
Lecture32:Limitcycles:existenceandnon-existencecriteria.
Lecture33:Relationbetweennon-linearsystemsandfirst-orderODE's;structuralstabilityofasystem,borderlinesketchingcases;illustrationsusingVolterra'sequationandprinciple.