高等數(shù)學(xué)第十二章常微分方程習(xí)題課ppt課件

1、第十二章第十二章 常微分方程常微分方程 習(xí)題課習(xí)題課:. 一階微分方程一可分離變量方程. 1)()(yxdxdy dxxydy)()( 齊次方程. 2 xyfdxdydxduxudxdyuxyxyu , 則令),(ufdxduxu ,)(uufdxdux .)( xdxuufdu線性方程. 3)()(xQyxpy cdxexQeydxxpdxxp)()()(公式伯努利方程. 4),()()(10 nyxQyxpyn)()(.xQyxpyynn 1解yynzyznn )(,11則令)()()()(xQnzxpnz 11全微分方程. 5xQyPdyyxQdxyxP 且0),(),( ),(),()

2、,(),(),(yxyxdyyxQdxyxPyxu00.),(為隱式通解cyxu ？”“積分因子尋找:是一種全微分方程可分離變量方程實際上.)()(01dyydxx .是分是合要靈活運(yùn)用與 dxdy)(.xfdxydnn 1:. 可降階方程二.次直接積分 n:. 殘缺二階方程2.),(xyxyyyF或中不顯含0 ),().xyfy 1pyxpy 則令),(.)(),(的一階方程xpxpfp :推廣)()(,1nnyxfy)(),()()(xpyxpynn 則令1),().yyfy 2ppyypy 則令,)(.)(),(的一階方程ypypfpp .然后好求解判別一階方程類型，:一階方程的常見形式

3、),(yxfdxdy 或0 dyyxQdxyxP),(),(?:式之一將待解方程化成下列形思考;可分離變量;齊次方程;線性方程;伯努利方程.全微分方程?積分因子?變量代換yexdydx2111)(.例1112)(.xexdydy解可分離變量112 xdxedyycxexy 21)(02dyxyxdxxyyxcoscos.例xyxxyyxdxdycoscos. 解xyxyxycoscos 1齊次方程xdudxudxdyxuyxyu ,則令,coscosuuuxdudxu1uuuuuxdudxcoscoscos11 ,cosuxdudx1 ,cosxdxduu .sinxycex 13 xyxdy

4、dxsin)(cos.例xyxxdydsec)(tan. 解一階線性方程cdxexeydxxdxxtantanseccdxexexxcoslncoslnseccdxxx2seccoscxxtancosxcxcossin 334yxxyxdyd .例)(.伯努利方程解323xyxxdydy xdydyzyz322 則令,322xxzz dxexcezxdxxdx2322dxexcexx22321222xecexx122 xcexyyyeeeyDD011 dxexx 232 dyyeyxyxdxdy22)(1 yey122xex21y )(隱式通解053223 dyyyxdxxyx)()(.例,.

5、3223yyxQxyxP 解,xQxyyP 2.原方程是全微分方程 ),(),(),(yxQdyPdxyxu00 ),(),()()(yxdyyyxdxxyx003223),( 00),(0 x),(yx yxdyyyxdxx032030)()(4224412141yyxx cyyxx 4224412141.為原方程的隱式通解.)()(.又解例053223 dyyyxdxxyx3223yyxxyxxdyd 33221xyxyxy 齊次方程.,xdudxuxdyduxyxyu 則設(shè),321uuuxdudxu ,uuuuuuxdudx2342121 ,xxduudu 21,lnln)ln(cxu

6、2121,lnln)ln(cxu2212 ,)(cux2122 .222cyx053223 dyyyxdxxyx)()(.例02222 dyyxydxyxx)()(,事實上022 )()(ydyxdxyx)(1022 yx)(20 ydyxdx或cyx 2221212)()(3222cyx 或.)()(中式已包含在此隱式解 31:,要熟悉幾個微分算式尋找積分因子)()(1xdyydxyxd)(42xydxxdyxyd)(52yxdyydxyxd)(ln6xyydxxdyxyd)(arctan722yxydxxdyxyd)()(ln2xyxdyydxxyd)(3122yxxdyydxxyd026

7、2 dxyxydyxdx.例022 dxyyxdxdx)(.解dxyyxdxdx22 )(222xdxyyxdxdx )(xydxd2)(ln.ln為原方程的隱式通解cxyx 2,)()sin(.與路徑無關(guān)已知例Ldyxxfdxxxyx1 . )(,)(xffxf求是可微函數(shù)且02 :雜例.)()(,sin2xxfxfxxQxxyP ,)(,sin.xxfQxxyxP 解,)()(sin2xxfxfxxx ,sin)()(xxxfxxf21 dxexxcexfdxxdxx121sin)(,sin)()(xxxfxxf21 dxxxcxsinxxxxDDsincossin 011xxxcxsin

8、cos ,102002cf 由.1 c.cossin)(xxxxxxf 2:)(.滿足可微函數(shù)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 1100200)()()(lim)()(lim)(xfxfxfxfxfxfxx 1

9、100200)()(xff210)(xfr21,)()(rxfxf 21,)()(dxrdxxfxf21,)(arctancxrxf ,)(arctancf 000 c,)(arctanxrxf .)tan()(xrxf #dxxyxydxxdy)(. 233例xdxyxyxd)()(.122 解,yxu 令,)(xdxudu12 則,xdxudu 12,xdxudu12cxuu 2211121lncxuu2112 ln)(.cxececxyxy222211 隱式通解1423 )()(.xyxxyxxdyd例,.1 xdydxdzdxyz則令解,023 zxzxxdzd23zxxzz 的伯努利

10、方程)(xzz ,312xzxzz ,zzuzu 21則令3xuxu dxxdxexcxdxeu3 22xe dxexcx232 22xeu dxexcx2322222 xecx tdetttx)(222dxexx232ttteeetDD0221 )(22 tet)(2222 xex,xyzu 11,21222 xecxyx21222 xecxyx#05 dyyxdxyyyx)cos()sin(.例0 dyydxyyxxdyydxcos)sin(.解dxyyxydyxd)sin(sin)( dxyyxyyxd sin)sin(yyxyxsin),( 1 cxyyxln)sinln( .sinx

11、ceyyx #yyxyyyxPsinsin ,sin yyxy 1,sincosyyxyxQ 2)sin()cos()sin(yyxyxyyyxyPxQ:檢驗)()()(.112113622 yxyxxdyd例,.11 yvxu令解)()(11 udvdxdyd則vuvuudvd2322 uvuv232齊次方程udzduzudvduzvuvz , 則令zzudzduz232 udvd 3427 yxyxxdyd.例)(不是齊次方程)()()()(.byaxbyaxxdyd 42解3042baba12ba)()()()(121422 yxyx,12 yvxu令)()(21 udvdxdyd則ud

12、vd vuvuudvd 42 uvuv142齊次方程:線性微分方程)()()()()()(10111 yxPyxPyxPynnnnnnycycycY 2211通解個線性無關(guān)的函數(shù)是 nyyyn,21),(常數(shù)線性無關(guān) kyyyy2121)()()()()()()(2111xQyxPyxPyxPynnnn *yycycycyYynn 2211通解;)()(的通解的對應(yīng)齊次方程是12Y.)(*的一個特解是 2y？疊加原理:常系數(shù)線性微分方程)()()(30111 ypypypynnnn)()()()(4111xfyPypypynnnn 特征方程)(50111 nnnnPrPrPr:)()(項之通解

13、的個根對應(yīng)式的nn35rk 重實根ik 重復(fù)根 xxcxccekkx cos)(1110 xxdxddkk sin)(1110 xrkkexcxcc)(1110 )()()()(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？重根式

14、的是)(5 kxmexPxf )()( )(xPemx 之特例1)(xPm0 xxP cos)(:類似地0 xxPexcos)(xe kxakx)(xQm xx sincos kx)()(xQ1)()(xQ2xxPn sin)( xx sincos kx)()(xQn1)()(xQn2i 此時特征根為xxPxxPn sin)(cos)(: 我們還有 xx sincos kx)()(xQm1)()(xQm2,maxnm 其中:例常系數(shù)線性微分方程舉)(.5331 xeyyyyx例013323 rrr:. 特征方程解,)(013 r1 r)( 三重實根,)(:xexcxccY 2210齊次通解,

15、)(*bxaexyx 3設(shè):,后得確定求導(dǎo)三次后代入原方程ba.*652413xexyx.*01223465241cxcxcxxeyYyx )(,)()cos(.20012214200 xxyyxxyy例,:.042 r特征方程解ir2 )(單根xcxcY2221sincos: 齊次通解0021exbxax)( x221cos022exdxcx)sincos( )sincos(*)(xdxcxbxay221 式的特解設(shè):,)(dcba后定出系數(shù)代入方程 1.,810081 dcba)sin(sin*xxxxxy21828181 :)(式的通解1)sin(sincos*xxxcxcyYy2182

16、221 .,)(1610221 cc由. )sin(sin*xxxy2182161 )(,)()cos(.20012214200 xxyyxxyy例xcxcY2221sincos )sin(sin*xxxxxy21828181 例雜假定開始下滑的鏈條自桌上無磨擦地長為例,.M61.,)(0100 vMt初速度鏈條垂下部分為時?部滑過桌子問需要多長時間鏈條全0X M1. 建立坐標(biāo)如圖解, )(txx 鏈條的位移函數(shù)為.,0000 ttxx則, 設(shè)鏈條的線密度為,)(gxf1 則鏈條受力. 6 m鏈條質(zhì)量,fxm )(16 xgx )(,)(20016600ttxxgxgx,)(gxx16 )(,

17、)(20016600ttxxgxgx:)(1先解微分方程,062 gr6gr ,tgtgececX6261 ax *設(shè)1 *x116261 ttggececxXx*)(之通解060122121)()(ccgcc由2121cc16ch1212166tgeextgtg16chtgx, 鏈條全部滑過桌子時當(dāng)5 x.,成立此時516chtg,ln1666char62tg.ln.356896t:)(.滿足可微函數(shù)例x 2,)()()( xxduuuxex0 . )(x 求 xxxdxuuduuxex00)()()(. 解)()()()(xxxxduuexxx 0 xxduue0)( )()(xexx 1010)(,)()()( xexx. )sin(cos)(xexxx 21 ,)(.在原點(diǎn)相切與曲線例2333xxyxfy ,)()()(xxxexftdtfxf 3320且有. )(xf求)()(21,