二阶变系数线性非齐次微分方程的通解公式

[]2007-04-23[](1970),,,,;(1979),,,,,(,100101)[]为了更多地得到理论上和应用上占有重要地位的二阶变系数线性非齐次微分方程的通解,这里使用常数变易法,在先求得二阶变系数线性齐次微分方程一个特解的情况下,将二阶变系数线性非齐次微分方程转化为可降阶的微分方程,从而给出了一种运算量较小的二阶变系数线性非齐次微分方程通解的一般公式,并且将通解公式进行了推广,实例证明该方法是可行的[]二阶变系数线性非齐次微分方程;通解;特解[]O175[]A[]1005-0310(2007)04-0074-03y+p(x)y+q(x)y=f(x)(1),,([1])(1)p(x),q(x)([2])r2+p(x)r+q(x)0(2)(rR),,(1)y=u(x)v(x)uvu=u(x),v=v(x),y=uv(1)y=uv+uv,y=uv+2uv+uvy,y,y(1)uv+(2u+p(x)u)v+(u+p(x)u+q(x)u)v=f(x)(3)u=u(x)(3)u+p(x)u+q(x)u=0(4)(4),[2]p(x),q(x)(2),(4)u=erx(5)v=v(x)(4)(3)v+2uu+p(x)v=f(x)u(6)(6)(6)(v=v(x))v=1u2e-p(x)dxf(x)uep(x)dxdx+c1dx+c2p(x)dx,f(x)uep(x)dxdx1u2e-p(x)dxf(x)uep(x)dxdx+c1dx,c1c2(1)y=erx1u2e-p(x)dxf(x)uep(x)dxdx+c1dx+c2,(1):(1),p(x),q(x),r(2),(1)y=erx1u2e-p(x)dxf(x)uep(x)dxdx+c1dx+c2(7)20071221470()JournalofBeijingUnionUniversity(NaturalSciences)Dec.2007Vol.21No.4SumNo.70p(x)dx,f(x)uep(x)dxdx1u2e-p(x)dxdxf(x)uep(x)dxdx+c1dx,c1c21y-1xy-9+3xy=xe3x:[2]r=-3,y-1xy-9+3xy=0u=e-3xp(x)=-1x,f(x)=xe3x,(7)y=e-3xe6xe1xdxxe3xe-3xe-1xdxdx+c1dx+c2=e-3xxe6x(dx+c1)dx+c2=e-3xxe6x(x+c1)dx+c2=e-3x16x2e6x+c16-118xe6x+1108-c136e6x+c2=16x2e3x+c16-118xe3x+1108-c136e3x+c2e-3x2y+2+1xy+1+1xy=xe-x:[2]r=-1,y+2+1xy+1+1xy=0u=e-xp(x)=2+1x,f(x)=xe-x,(7)y=e-xe2xe-2+1xdxxe-xe-xe2+1xdxdx+c1dx+c2=e-x1x(x2dx+c1)dx+c2=e-x13x2+c1xdx+c2=19x3e-x+c1e-xln|x|+c2e-x:r2+p(x)r+q(x)0,,,(1)u=u(x),(7)(1)Liouville([3])3y+2xy+y=1xcosx:y+2xy+y=0u=sinxxp(x)=2x,f(x)=1xcosx,(7)y=sinxxx2sin2xe-2xdxcosxxsinxxe2xdxdx+c1dx+c2=sinxx1sin2x(sinxcosxdx+c1)dx+c2=sinxx12+c1csc2xdx+c2=sinxx12x-c1cotx+c2=12sinx-c1xcosx+c2xsinx[][1].[J].,2002,5(2):10-13.[2],.[J].,2006,9(3):22-24.[3],.[M].2.:,1983.[4].[J].:,2007,21(8):136-141.[5],,.[J].,2003,17(6):38-39.75214:TheFormulaofGeneralSolutiontoSecondOrderLinearDifferentialEquationwithVariableCoefficientsXINGChun-feng,YUANAn-feng(BasicCoursesDepartmentofBeijingUnionUniversity,Beijing100101,China)Abstract:Inordertoobtainmoregeneralsolutiontosecondorderlineardifferentialequationwithvariablecoefficientswhichisimportantintheoryandpractice,onthebasisofknowingaspecialsolutionofthesecondorderlineardifferentialequationwithvariablecoefficientsandbyusingthemethodofvariationofconstant,thesecondorderlineardifferentialequationwithvariablecoefficientsistransferredtothereduceddifferentialequationandageneralformulaofthesecondorderlineardifferentialequationwithvariablecoefficientsisderived.Examplesaregiventoverifythemethod.Keywords:secondorderlineardifferentialequationwithvariablecoefficients;generalsolution;particularsolution(责任编辑)(上接第70页)TheConvergenceofQuasi-NewtonMethodsforUnconstrainedOptimizationWANGL-iwei,LIUDa-lian(BasicCoursesDepartmentofBeijingUnionUniversity,Beijing100101,China)Abstract:AclassofalgorithmswhichareupdateQuas-iNewtonmethodsforunconstrainedoptimizationareasfollows:minf(x),xRn.Theproofoftheglobalandsuperlinearlyconvergenceofthegeneralized-quas-iNewtonmethordsisproposed.Threemaintheoremsaregiven.Keywords:generalized-quas-iNewtonalgorithms;superlinearlyconvergence;uncontrainedoptimization(责任编辑)76()200712