常微分方程专题选讲小题目

《常微分方程专题选讲小题目》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、PROBLEMSINTOPICSOFORDINARYDIFFERENTIALEQUATIONSBYJIANGANGQITopic1.ExactnessofsecondorderdifferentialequationsP11:Provethatf(x;y′;y′′)=0isexactifandonlyif()()@f@fDxDx=0@y′′@y′P12:Provethatf(y;y′;y′′)=0isexactifandonlyif()()@f@f@fDx+Dx=0@y@y′@y′′P13:Provethatf(x;y;y′′)=0isexactifandonlyif()@f@f+Dx=

2、0@y@y′′P14:Provethaty′′=f(x;y;y′)0isexactifandonlyif()@f@fDx=0@y@y′P15:Findoutthesufficientandnecessaryconditionfortheequation′′′f(x;y)+g(y;y)=0tobeexactandgivetheproof.P16:Findoutthesufficientandnecessaryconditionfortheequation′′′f(x;y)+g(y;y)=0tobeexactandgivetheproof.P17:Findoutthesufficientandneces

3、saryconditionfortheequation′′′f(x;y)+g(y;y)=0tobeexactandgivetheproof.P18:Constructanexactsecondorderdifferentialequationandsolveit(Four).P18:Constructanexactsecondorderdifferentialequationandsolveit(Four).P18:Constructanexactsecondorderdifferentialequationandsolveit(Four).12P18:Constructanexactsecon

4、dorderdifferentialequationandsolveit(Four).P19:Studyintegralfactorsofnon-exactsecondorderdifferentialequations(Two).P19:Studyintegralfactorsofnon-exactsecondorderdifferentialequations(Two).Topic3.Loweranduppersolutions,monotoneiterativemethodsforIVPoffirstorderdifferentialequationsP31:ConsidertheIVPoffi

5、rstorderdifferentialequations′u=f(t;u);u(0)=u0;t2[0;T]:(IVP+)Letfsatisfyone-sideLipschitzconditionf(t;u1)f(t;u2)L(u1u2)wheneveru1u2forsomeL>0.Provetheuniquenessofsolutionsof(IVP+).(Two)P31:ConsidertheIVPoffirstorderdifferentialequations′u=f(t;u);u(0)=u0;t2[0;T]:(IVP+)Letfsatisfyone-sideLipschitzc

6、onditionf(t;u1)f(t;u2)L(u1u2)wheneveru1u2forsomeL>0.Provetheuniquenessofsolutionsof(IVP+).(Two)P32:ConsidertheIVPoffirstorderdifferentialequations′u=f(t;u);u(0)=u0;t2[T;0]:(IVP)Letfsatisfyone-sideLipschitzconditionf(t;u1)f(t;u2)L(u1u2)wheneveru1u2forsomeL>0.Provetheuniquenessofsolutionsof

7、(IVP-).(Two)P32:ConsidertheIVPoffirstorderdifferentialequations′u=f(t;u);u(0)=u0;t2[T;0]:(IVP)Letfsatisfyone-sideLipschitzconditionf(t;u1)f(t;u2)L(u1u2)wheneveru1u2forsomeL>0.Provetheuniquenessofsolutionsof(