高等数学第十二章常微分方程习题课

1第十二章常微分方程习题课:.一阶微分方程一可分离变量方程.1)()(yxdxdydxxydy)()(齐次方程.2xyfdxdydxduxudxdyuxyxyu,,则令),(ufdxduxu,)(uufdxdux.)(xdxuufdu2线性方程.3)()(xQyxpycdxexQeydxxpdxxp)()()(公式伯努利方程.4),()()(10nyxQyxpyn)()(.xQyxpyynn1解yynzyznn)(,11则令)()()()(xQnzxpnz11全微分方程.5xQyPdyyxQdxyxP且0),(),(),(),(),(),(),(yxyxdyyxQdxyxPyxu00.),(为隐式通解cyxu？”“积分因子寻找3:是一种全微分方程可分离变量方程实际上.)()(01dyydxx.是分是合要灵活运用与dxdy)(.xfdxydnn1:.可降阶方程二.次直接积分n:.残缺二阶方程2.),,,(xyxyyyF或中不显含0),().xyfy1pyxpy则令),(.)(),(的一阶方程xpxpfp:推广)()(,1nnyxfy)(),()()(xpyxpynn则令14),().yyfy2ppyypy则令,)(.)(),(的一阶方程ypypfpp.然后好求解判别一阶方程类型，:一阶方程的常见形式),(yxfdxdy或0dyyxQdxyxP),(),(?:式之一将待解方程化成下列形思考;可分离变量;齐次方程;线性方程;伯努利方程.全微分方程?积分因子?变量代换5yexdydx2111)(.例1112)(.xexdydy解可分离变量112xdxedyycxexy21)(02dyxyxdxxyyxcoscos.例xyxxyyxdxdycoscos.解xyxyxycoscos1齐次方程xdudxudxdyxuyxyu,,则令,coscosuuuxdudxu1uuuuuxdudxcoscoscos116,cosuxdudx1,cosxdxduu.sinxycex13xyxdydxsin)(cos.例xyxxdydsec)(tan.解一阶线性方程cdxexeydxxdxxtantanseccdxexexxcoslncoslnseccdxxx2seccoscxxtancosxcxcossin7334yxxyxdyd.例)(.伯努利方程解323xyxxdydyxdydyzyz322则令,322xxzzdxexcezxdxxdx2322dxexcexx22321222xecexx122xcexyyyeeeyDD011dxexx232dyyeyxyxdxdy22)(1yey122xex21y)(隐式通解8053223dyyyxdxxyx)()(.例,,.3223yyxQxyxP解,xQxyyP2.原方程是全微分方程),(),(),(yxQdyPdxyxu00),(),()()(yxdyyyxdxxyx003223),(00),(0x),(yxyxdyyyxdxx032030)()(4224412141yyxxcyyxx4224412141.为原方程的隐式通解9.)()(.又解例053223dyyyxdxxyx3223yyxxyxxdyd33221xyxyxy齐次方程.,,xdudxuxdyduxyxyu则设,321uuuxdudxu,uuuuuuxdudx2342121,xxduudu21,lnln)ln(cxu2121,lnln)ln(cxu2212,)(cux2122.222cyx10053223dyyyxdxxyx)()(.例02222dyyxydxyxx)()(,事实上022)()(ydyxdxyx)(1022yx)(20ydyxdx或cyx2221212)()(3222cyx或.)()(中式已包含在此隐式解3111:,要熟悉几个微分算式寻找积分因子)()(1xdyydxyxd)(42xydxxdyxyd)(52yxdyydxyxd)(ln6xyydxxdyxyd)(arctan722yxydxxdyxyd)()(ln2xyxdyydxxyd)(3122yxxdyydxxyd120262dxyxydyxdx.例022dxyyxdxdx)(.解dxyyxdxdx22)(222xdxyyxdxdx)(xydxd2)(ln.ln为原方程的隐式通解cxyx213,)()sin(.与路径无关已知例Ldyxxfdxxxyx1.)(,)(xffxf求是可微函数且02:杂例.)()(,sin2xxfxfxxQxxyP,)(,sin.xxfQxxyxP解,)()(sin2xxfxfxxx,sin)()(xxxfxxf2114dxexxcexfdxxdxx121sin)(,sin)()(xxxfxxf21dxxxcxsinxxxxDDsincossin011xxxcxsincos,102002cf由.1c.cossin)(xxxxxxf215:)(.满足可微函数例xf2)()()()()(yfxfyfxfyxf1.)(,)()(xfrrf求已知且0)()()()()()()(.xfxfxfxfxfxfxxf1解)()()()()(xfxfxfxfxf12xxfxxfxfx)()(lim)(0)()()()(limxfxfxfxxfx1120)()()()()(0010000fffff)()(唯一00f)()()(lim)()(limxfxfxfxfxfxx110020016)()()(lim)()(lim)(xfxfxfxfxfxfxx1100200)()(xff210)(xfr21,)()(rxfxf21,)()(dxrdxxfxf21,)(arctancxrxf,)(arctancf000c,)(arctanxrxf.)tan()(xrxf#17dxxyxydxxdy)(.233例xdxyxyxd)()(.122解,yxu令,)(xdxudu12则,xdxudu12,xdxudu12cxuu2211121lncxuu2112ln)(.cxececxyxy222211隐式通解181423)()(.xyxxyxxdyd例,,.1xdydxdzdxyz则令解,023zxzxxdzd23zxxzz的伯努利方程)(xzz,312xzxzz,,zzuzu21则令3xuxudxxdxexcxdxeu322xedxexcx2321922xeudxexcx2322222xecxtdetttx)(222dxexx232ttteeetDD0221)(22tet)(2222xex,xyzu11,21222xecxyx21222xecxyx#2005dyyxdxyyyx)cos()sin(.例0dyydxyyxxdyydxcos)sin(.解dxyyxydyxd)sin(sin)(dxyyxyyxdsin)sin(yyxyxsin),(1cxyyxln)sinln(.sinxceyyx#yyxyyyxPsinsin,sinyyxy1,sincosyyxyxQ2)sin()cos()sin(yyxyxyyyxyPxQ:检验21))(()()(.112113622yxyxxdyd例,,.11yvxu令解)()(11udvdxdyd则vuvuudvd2322uvuv232齐次方程udzduzudvduzvuvz,,则令zzudzduz232udvd223427yxyxxdyd.例)(不是齐次方程)()()()(.byaxbyaxxdyd42解3042baba12ba)()()()(121422yxyx,,12yvxu令)()(21udvdxdyd则udvdvuvuudvd42uvuv142齐次方程23:线性微分方程)()()()()()(10111yxPyxPyxPynnnnnnycycycY2211通解个线性无关的函数是nyyyn,,,,21),(常数线性无关kyyyy2121)()()()()()()(2111xQyxPyxPyxPynnnn**yycycycyYynn2211通解;)()(的通解的对应齐次方程是12Y.)(*的一个特解是2y？叠加原理24:常系数线性微分方程)()()(30111ypypypynnnn)()()()(4111xfyPypypynnnn特征方程)(50111nnnnPrPrPr:)()(项之通解的个根对应式的nn35rk重实根ik重复根xxcxccekkxcos)(1110xxdxddkksin)(1110xrkkexcxcc)(111025)()()()(4111xfyPypypynnnn特征方程)(50111nnnnPrPrPrxmexPxf)()()1xey*kxxxPexfxcos)()()2xxeyxsincos*kx)()(xQ1)()(xQ2xxPexfnxsin)()()3xxeyxsincos*kx)()(xQn1)()(xQn2)(xQmxxPxxPexfnxsin)(cos)()()4重根式的是)(5ik？重根式的是)(5k26xmexPxf)()()(xPemx之特例1)(xPm0xxPcos)(:类似地0xxPexcos)(xekxakx)(xQmxxsincoskx)()(xQ1)()(xQ2xxPnsin)(xxsincoskx)()(xQn1)()(xQn2i此时特征根为xxPxxPns