常微分方程初步 分离变量法 (英文)

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1、SeparableEquationsSection2.2inBoyceandDiPrimaWenhanWangDepartmentofMathematicsUniversityofWashington10-01-2012WenhanWangDepartmentofMathematicsUniversityofWashington()FirstOrderODE10-01-20121/13SeparableFirstOrderEquationsWeconsideranotherclassoffirstorderequations.Thegeneralformoffirs

2、torderequationisdy=f(t;y):dtAnysuchequationcanbewrittenasdyh1(t;y)+h2(t;y)=0:dtIfithappensthath1(t;y)isafunctiononlyintandh2(t;y)isafunctiononlyiny,thendyh1(t)+h2(y)=0:dtSuchequationsarecalledseparableequations.WenhanWangDepartmentofMathematicsUniversityofWashington()FirstOrderODE10-

3、01-20122/13SeparableFirstOrderEquationsExampleTheequationdyt2=dt1y2isseparable.MultiplybythedenominatorofRHS:2dy2(1y)=tdtMoveeverythingtotheRHS:22dyt+(y1)=0dtSeth1(t)=t2,h2(y)=y21.WenhanWangDepartmentofMathematicsUniversityofWashington()FirstOrderODE10-01-20123/13HowtoSolveSepara

4、bleSolutions:DirectIntegrationSupposewehavethegeneralseparableequationdyh1(t)+h2(y)=0:dtIntegrateh1(t)w.r.t.t,wegetZh1(t)dt:=H1(t):dyIntegrateh2(y)dtw.r.t.t,wegetfromchainrule:ZZdyh2(y)dt=h2(y)dy:=H2(y):dtThedifferentialequationbecomesH1(t)+H2(y)=c:WenhanWangDepartmentofMathematicsUn

5、iversityofWashington()FirstOrderODE10-01-20124/13DirectIntegrationTheoremSupposey=y(t)satisfiestheseparableequationdyh1(t)+h2(y)=0;dtordifferentialformh1(t)dt+h2(y)dy=0:LetH1(t)beananti-derivativeofh1(t)andH2(y)beananti-derivativeofH2(y).ThenZZh1(t)dt+h2(y)dy=H1(t)+H2(y)=c:Thisiscalle

6、dtheimplicitgeneralsolution.WenhanWangDepartmentofMathematicsUniversityofWashington()FirstOrderODE10-01-20125/13Precautions!Beforeapplyingthemethodofdirectintegration,makesurethattheequationisindeedseparable.Transformtheequationintothestandardformdyh1(t)+h2(y)=0orh1(t)dt+h2(y)dy=0dtT

7、hisiscalledseparationofvariables.Theconstantdoesnotmatterfortheanti-derivativesH1(t)andH2(y),butdon’tforgettheconstantontheRHS.Itisnotalwayspossibletosolveforanexplicitformy=y(t)fromtheimplicitsolutionH1(t)+H2(y)=c.ForanIVP,plugintheinitialconditiontofindtheconstantc.WenhanWangDepartm

8、entofMathema