1、1第十二章第十二章 常微分方程常微分方程 习题课习题课:.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),(),(),(),(),(),(),(
2、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.)()(01dyydxx .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)()(,1nnyxfy)(),()()(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),(yxfdxd
3、y 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5yexdydx2111)(.例1112)(.xexdydy解可分离变量112 xdxedyycxexy 21)(02dyxyxdxxyyxcoscos.例xyxxyyxdxdycoscos.解xyxyxycoscos 1齐次方程xdudxudxdyxuyxyu ,则令,coscosuuuxdudxu1uuuuuxdudxcoscoscos11 6,cosuxdudx1,cosxdxduu.sinxycex 13 xyxdydxsin)(cos
4、.例xyxxdydsec)(tan.解一阶线性方程cdxexeydxxdxxtantanseccdxexexxcoslncoslnseccdxxx2seccoscxxtancosxcxcossin 7334yxxyxdyd .例)(.伯努利方程解323xyxxdydy xdydyzyz322 则令,322xxzz dxexcezxdxxdx2322dxexcexx22321222xecexx122 xcexyyyeeeyDD011 dxexx 232 dyyeyxyxdxdy22)(1 yey122xex21y)(隐式通解8053223 dyyyxdxxyx)()(.例,.3223yyxQxy
5、xP 解,xQxyyP 2.原方程是全微分方程 ),(),(),(yxQdyPdxyxu00 ),(),()()(yxdyyyxdxxyx003223),(00),(0 x),(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)
6、ln(cxu2212 ,)(cux2122 .222cyx10053223 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)(3122yxxdyydxxyd120262 dx
7、yxydyxdx.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 14dxexxcexfdxxdxx121sin)(,sin)()(xxxfxxf21 dxxxcxsinxxxxDDsincossin 011xxxcxsincos
8、,102002cf 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 110
9、0200)()(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
10、程)(xzz,312xzxzz ,zzuzu 21则令3xuxu dxxdxexcxdxeu3 22xe dxexcx23219 22xeu dxexcx2322222 xecx tdetttx)(222dxexx232ttteeetDD0221 )(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(.sinxc
11、eyyx#yyxyyyxPsinsin ,sin yyxy 1,sincosyyxyxQ 2)sin()cos()sin(yyxyxyyyxyPxQ: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3042baba12ba)()()()(121422 yxyx,12 yvxu令)()(21 udvdxdyd则u
12、dvd vuvuudvd 42 uvuv142齐次方程23:线性微分方程)()()()()()(10111 yxPyxPyxPynnnnnnycycycY 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:)()
13、(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 nnnnPrPrPrxmexPxf)()()1xey *kxxxPexfx cos)()()2 xxeyx sincos*kx)()(xQ1)()(xQ2xxPexfnx sin)()()3 xxeyx sincos*kx)()(xQn1)()(xQn2)(xQm xxPxxPexfnx sin)(cos)()()4重根式的是)(5 i k？重根
14、式的是)(5 k26xmexPxf)()()(xPemx 之特例1)(xPm0 xxP cos)(:类似地0 xxPexcos)(xe kxakx)(xQm xx sincos kx)()(xQ1)()(xQ2xxPn sin)(xx sincos kx)()(xQn1)()(xQn2i 此时特征根为xxPxxPn sin)(cos)(:我们还有 xx sincos kx)()(xQm1)()(xQm2,maxnm 其中27:例常系数线性微分方程举)(.5331 xeyyyyx例013323 rrr:.特征方程解,)(013 r1 r)(三重实根,)(:xexcxccY 2210齐次通解,)(
15、*bxaexyx 3设:,后得确定求导三次后代入原方程ba.*652413xexyx.*01223465241cxcxcxxeyYyx28)(,)()cos(.20012214200 xxyyxxyy例,:.042 r特征方程解ir2 )(单根xcxcY2221sincos:齐次通解0021exbxax)(x221cos022exdxcx)sincos()sincos(*)(xdxcxbxay221 式的特解设:,)(dcba后定出系数代入方程 1.,810081 dcba)sin(sin*xxxxxy21828181 29:)(式的通解1)sin(sincos*xxxcxcyYy218222
16、1 .,)(1610221 cc由.)sin(sin*xxxy2182161 )(,)()cos(.20012214200 xxyyxxyy例xcxcY2221sincos )sin(sin*xxxxxy21828181 30例杂假定开始下滑的链条自桌上无磨擦地长为例,.M61.,)(0100 vMt初速度链条垂下部分为时?部滑过桌子问需要多长时间链条全0X M1.建立坐标如图解,)(txx 链条的位移函数为.,0000 ttxx则,设链条的线密度为,)(gxf1 则链条受力.6 m链条质量,fxm )(16 xgx)(,)(20016600ttxxgxgx,)(gxx16 31)(,)(20
17、016600ttxxgxgx:)(1先解微分方程,062 gr6gr ,tgtgececX6261 ax *设1 *x116261 ttggececxXx*)(之通解060122121)()(ccgcc由2121cc16ch1212166tgeextgtg3216chtgx,链条全部滑过桌子时当5 x.,成立此时516chtg,ln1666char62tg.ln.356896t33:)(.满足可微函数例x 2,)()()(xxduuuxex0 .)(x 求 xxxdxuuduuxex00)()()(.解)()()()(xxxxduuexxx 0 xxduue0)()()(xexx 1010)(
18、,)()()(xexx.)sin(cos)(xexxx 21 34,)(.在原点相切与曲线例2333xxyxfy ,)()()(xxxexftdtfxf 3320且有.)(xf求)()(21,)(:)(.001 f知由解;)(030023xxxfxxxeexfxfxf 33322)()()()(由)()()()(3323 xexfxfxfx,)(,)()()()()(00003323ffxexfxfxfx35)()(,)()()()()()(4000033323ffxexfxfxfx,0232 rr.,2121 rr,)(xxececxF221 )()(*bxaexfx 设x.,623 ba)(*)()()(xfxFxf 之通解3xxeececxxx6232221.,)(66421 cc由.)(xxeeexfxxx6236622
