常微分方程初步 二阶微分方程 特征方程 (英文)

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1、IntroductiontoSecondOrderODEsSection3.1inBoyceandDiPrimaWenhanWangDepartmentofMathematicsUniversityofWashington10-15-2012WenhanWangDepartmentofMathematicsUniversityofWashington()SecondOrderODE10-15-20121/15SecondOrderEquationsandVibrationModelsConsideraspring-masssystem.Theobjectwithmassmisatta

2、chedtoaspringwithspringconstantk.Lety(t)bethedisplacementoftheobjectfromtheequilibriumposition.Assumealsothatthereisaresistanceforceproportionaltothedyvelocity.dtMoreover,thereisanexternalforceF(t).ByNewton’sSecondLaw,wehaved2ydym++ky=F(t)dt2dtWenhanWangDepartmentofMathematicsUniversityofWashin

3、gton()SecondOrderODE10-15-20122/15ClassificationofSecondOrderEquationsThegeneralformofsecondorderequationisd2ydy=ft;y;:dt2dtWesaythatthesecondorderequationislinear,ifd2ydy=g(t)p(t)q(t)y:dt2dtAnotherformisP(t)y00+Q(t)y0+R(t)y=G(t):IfG(t)=0,wecalltheequationhomogeneous.WenhanWangDepartmentofMa

4、thematicsUniversityofWashington()SecondOrderODE10-15-20123/15InitialConditionsforSecondOrderEquationsWehaveseenthatforafirstorderdifferentialequation,thereareinfinitelymanygeneralsolutions.Weneedaninitialconditionforfirstorderequationstodetermineauniquespecificsolution.Forsecondorderequations,wenee

5、dtwoinitialconditions.Theinitialconditionslookslikey(0)=y;y0(0)=y0:00Isthisphysicallyreasonable?Themotionofaspring-masssystemdependsonboththeinitialpositionandtheinitialvelocity.WenhanWangDepartmentofMathematicsUniversityofWashington()SecondOrderODE10-15-20124/15LinearSecondOrderEquationswithCo

6、nstantCoefficientsmy00+y0+ky=F(t)Thisequationislinearandofsecondorder,withconstantcoefficients.Thegeneralformforlinearsecondorderequationswithconstantcoefficientsisay00+by0+cy=f(t);(a6=0):Iff(t)=0,theequationbecomeshomogeneous:ay00+by0+cy=0;(a6=0)Thisisthegeneralformof(linearsecondorder)homogeneou

7、sequationswithconstantcoefficients.WenhanWangDepartmentofMathematicsUniversityofWashington()SecondOrderODE10-15-20125/15PrincipalofSuperpositionTheorem(PrincipalofSuperposition)GivenasecondorderhomogeneouslinearequationP(t)y00+Q(t)