东北师范大学《常微分方程》1部分习题解答.pdf 19页

'ȶڋၴ࠶ࠝڴȷრ࿸੶ܶՊͺಝӖ༰ฑדё߽ཙ೉ԛ͑催ㄝᬭ㚆ߎ⠜⼒1 д乬1.21∖ϟ߫ৃߚ⾏ব䞣ᖂߚᮍ⿟ⱘ䗮㾷˖(1)ydy=xdx121222㾷˖⿃ߚˈᕫy=x+cेx-y=c122dy(2)=ylnydxdy㾷˖y=0,y=1Ў⡍㾷ˈᔧy¹0,y¹1ᯊˈ=dxˈylnyx⿃ߚˈᕫlnlny=x+c,lny=±ec1ex=cexc¹0ˈेy=ece1dyx-y(3)=edxyxyx㾷˖বᔶᕫedy=edx⿃ߚˈᕫe-e=c(4)tanydx-cotxdy=0dytanycosysinx㾷˖বᔶᕫ=ˈy=0Ў⡍㾷ˈᔧy¹0ᯊˈdy=dx.dxcotxsinycosx⿃ߚˈᕫlnsiny=-lncosx+c,lnsinycosx=c,11ेsinycosx=±ec1=c,c¹02ˊ∖ϟ߫ᮍ⿟⒵䎇㒭ᅮ߱ؐᴵӊⱘ㾷˖dy(1)=y(y-1),y(0)=1dx11㾷˖y=0,y=1Ў⡍㾷ˈᔧy¹0,y¹1ᯊˈ(-)dy=dxˈy-1yy-1y-1c1xx⿃ߚˈᕫln=x+c,=±ee=ce,c¹01yyᇚy(0)=1ҷܹˈᕫc=0ˈेy=1Ў᠔∖ⱘ㾷DŽ22(2)(x-1)y¢+2xy=0,y(0)=12dy2xydy2x㾷˖=-,y=0Ў⡍㾷ˈᔧy¹0ᯊˈ=-dxˈ222dxx-1yx-12 12⿃ߚˈᕫ-=-lnx-1+cy1ᇚy(0)=1ҷܹˈᕫc=-1ˈेy=Ў᠔∖ⱘ㾷DŽ2lnx-1+1(3)y¢=33y2,y(2)=0dy㾷˖y=0Ў⡍㾷ˈᔧy¹0ᯊˈ=dxˈ23y3133⿃ߚˈᕫy=x+c,y=(x+c)3ᇚy(2)=0ҷܹˈᕫc=-2ˈेy=(x-2)੠y=0ഛЎ᠔∖ⱘ㾷DŽ2222(4)(y+xy)dx-(x+yx)dy=0,y(1)=-11+x1+y㾷˖x=0,y=0Ў⡍㾷ˈᔧx¹0,y¹0ᯊˈdx-dy=0ˈ22xy111111xcx-yx-y⿃ߚˈᕫ-+lnx+-lny=c,=±e1e=ce,c¹01xyy11x--2-2xyᇚy(1)=-1ҷܹˈᕫc=-eˈे=-eeЎ᠔∖ⱘ㾷DŽy224ˊ∖㾷ᮍ⿟x1-ydx+y1-xdy=0㾷˖x=±1(-1£y£1),y=±1(-1£x£1)Ў⡍㾷ˈxyᔧx¹±1,y¹±1ᯊˈdx+dy=0221-x1-y22⿃ߚˈᕫ1-x+1-y=c(c>0)6ˊ∖ϔ᳆㒓ˈՓ݊݋᳝ҹϟᗻ䋼˖᳆㒓Ϟ৘⚍໘ⱘߛ㒓Ϣߛ⚍ࠄॳ⚍ⱘ৥ᕘঞx䕈ৃೈ៤ϔϾㄝ㝄ϝ㾦ᔶ(ҹx䕈Ўᑩ)ˈϨ䗮䖛⚍(1,2).㾷˖䆒᠔∖᳆㒓Ўy=y(x)ᇍ݊Ϟӏϔ⚍(x,y)ⱘߛ㒓ᮍ⿟˖yY-y=y"(X-x)Ѣx䕈Ϟⱘ៾䎱Ўa=x-⬅乬ᛣᓎゟᮍ⿟˖y"3 yyx--x=x-0ेy"=-,y(1)=2y"xcc∖ᕫᮍ⿟ⱘ䗮㾷Ўxy=e,c¹0ݡ⬅2=eᕫc=ln2,ᕫ᠔∖᳆㒓ЎЎxy=27ˊҎᎹ㐕Ⅺ㒚㦠ˈ݊๲䭓䗳ᑺ੠ᔧᯊⱘ㒚㦠᭄៤ℷ↨˄1˅བᵰ4ᇣᯊⱘ㒚㦠᭄Ўॳ㒚㦠᭄ⱘ2סˈ䙷М㒣䖛12ᇣᯊᑨ᳝໮ᇥ˛44˄2˅བᵰ೼3ᇣᯊᯊⱘ㒚㦠᭄Ўᕫ10Ͼˈ೼5ᇣᯊᯊⱘ㒚㦠᭄Ўᕫ4´10Ͼˈ䙷М೼ᓔྟᯊ᳝໮ᇥϾ㒚㦠˛dq㾷˖䆒tᯊࠏⱘ㒚㦠᭄Ўq(t),⬅乬ᛣᓎゟᖂߚᮍ⿟=kqk>0dtktkt∖㾷ᮍ⿟ᕫq=ceݡ䆒t=0ᯊˈ㒚㦠᭄Ўq,∖ᕫᮍ⿟ⱘ㾷Ўq=qe004kln2˄1˅⬅q(4)=2qेqe=2qᕫk=0004ln212q(12)=qe12k=qe4=8q0003k45k4˄2˅⬅ᴵӊq(3)=qe=10,q(5)=qe=4´1000ln4ln43k34↨䕗ϸᓣᕫk=ˈݡ⬅q(3)=qe=qe2=8q=1000023ᕫq=1.25´100д乬1.31㾷ϟ߫ᮍ⿟˖22(2)(y-2xy)dx+xdy=0dyyy2㾷˖ᮍ⿟ᬍݭЎ=2()-()dxxxydu211dxҸu=ˈ᳝u+x=2u-uᭈ⧚Ў(-)du=(u¹0,1)xdxuu-1xuc1x⿃ߚˈᕫln=lncxेu=1u-1cx-11ҷಲব䞣ˈᕫ䗮㾷x(y-x)=cy,y=0гᰃᮍ⿟ⱘ㾷4 y(4)xy¢-y=xtanxdyyy㾷˖ᮍ⿟ᬍݭЎ-=tandxxxydusinudxҸu=ˈ᳝x=tanu=ेcotudu=(sinu¹0)xdxcosux⿃ߚˈᕫsinu=cxyҷಲব䞣ˈᕫ䗮㾷sin=cxxx+y(5)xy¢-y=(x+y)lnxdyyyx+y㾷˖ᮍ⿟ᬍݭЎ-=(1+)lndxxxxyduҸu=ˈ᳝x=(1+u)ln(1+u)xdxdudxᔧu¹0,u¹-1ᯊ=(1+u)ln(1+u)x⿃ߚˈᕫln(1+u)=cxyҷಲব䞣ˈᕫ䗮㾷ln(1+)=cxx22(6)xy¢=x-y+ydyy2y㾷˖ᮍ⿟ᬍݭЎ=1-()+dxxxydu2dudxҸu=ˈ᳝x=1-uߚ⾏ব䞣=(-10,ᇍܙߚ໻ⱘx,ᔧx>xᯊ,᳝|f(x)|0ᯊ,݊⿃ߚ᳆㒓བ೒(3)᠔⼎;ᔧy<0ᯊ,݊⿃ߚ᳆㒓བ೒(4)᠔⼎.12 ೒(3)೒(4)121(4)⬅Ѣf(x,y)=-,ϡձ䌪Ѣy,᠔ҹ,ৃⶹ೼Ⳉ㒓x=-Ϟ2xk㒓㋴എⱘ㒓㋴䛑ᑇ㸠,݊᭰⥛Ўেッߑ᭄f(x,y)῾തᷛᑇᮍⱘצ᭄ⱘⳌড᭄.Ѣᰃ,῾തᷛ䍞໻,㒓㋴എⱘᮍ৥䍞ᑇ㓧.Ң㗠,ৃҹḍ᥂㒓㋴എ㒓㋴ⱘ䍟࢓,໻ԧϞᦣߎ⿃ߚ᳆㒓.བ೒(5)᠔⼎.(5)⬅Ѣf(x,y)=x೒,(5)ϡձ䌪Ѣy,಴㗠೼Ⳉ㒓x=k(kЎᐌ᭄)Ϟ,㒓㋴എⱘ㒓㋴䛑ᑇ㸠,ᬙᔧx>0ᯊ,݊⿃ߚ᳆㒓བ೒(6)᠔⼎;ᔧx<0ᯊ,݊⿃ߚ᳆㒓བ೒(7)᠔⼎.೒(7)೒(6)2.䆩⬏ߎᮍ⿟dy22=x-ydx೼xoyᑇ䴶Ϟⱘ⿃ߚ᳆㒓ⱘ໻㟈೒ڣ.㾷˖䖭Ͼᮍ⿟ᰃϡৃ⿃ⱘ,ԚᯧѢ⬏ߎᅗⱘ㒓㋴എ.೼ৠϔҹॳ⚍Ўᇍ⿄Ёᖗⱘঠ᳆㒓Ϟ,㒓㋴എⱘ㒓㋴䛑ᑇ㸠.݊᭰⥛ㄝѢঠ᳆㒓ᅲञ䕈䭓ⱘᑇᮍ.Ѣᰃ,ᅲञ䕈䍞䭓,㒓㋴എⱘᮍ৥䍞䰵.Ң㗠,13೒(8) ḍ᥂㒓㋴എ㒓㋴ⱘ䍟࢓,໻ԧϞৃҹᦣߎ⿃ߚ᳆㒓.བ೒(8)᠔⼎.3.䆩⫼⃻ᢝᡬ㒓⊩,পℹ䭓h=0.1,∖߱ؐ䯂乬dy22=x+y,dx
y(1)=1ⱘ㾷೼x=1.4ᯊⱘ䖥Ԑؐ.㾷Ҹx=1,y=1.00߭x=x+0.1=1.1,y=1+2×0.1=1.2;101x=x+0.1=1.2,y=1.2+2.65×0.1=1.465;212x=x+0.1=1.3,y=1.465+3.586×0.1=1.824;323x=x+0.1=1.4,y=1.824+5.017×0.1=2.326.434д乬2.2dy1.䆩߸ᮁᮍ⿟=xtanx೼ऎඳdx(1)R:-1£x£1,0£y£p;1pp(2)R:-1£x£1,-£y£244Ϟᰃ৺⒵䎇ᅮ⧚2.2ⱘᴵӊ?p㾷˖(1)ϡ⒵䎇.಴Ў೼ऎඳRϞ,েッߑ᭄f(x,y)=xtanyᔧy=ᯊϡ䖲12㓁.(2)⒵䎇.಴Ў೼ऎඳRϞ,েッߑ᭄f(x,y)=xtany䖲㓁Ϩ2xf¢(x,y)=£2᳝⬠.y2cosy2.߸ᮁϟ߫ᮍ⿟೼ҔМḋⱘऎඳϞ䆕߱ؐ㾷ᄬ೼Ϩଃϔ?22(1)y¢=x+y;(2)y¢=x+siny;14 1-(3)y¢=x3;(4)y¢=y.22㾷˖(1)಴Ўf(x,y)=x+yঞf¢(x,y)=2y೼ᭈϾxoyᑇ䴶Ϟ䖲㓁,᠔ҹ೼yᭈϾxoyᑇ䴶Ϟ⒵䎇ᄬ೼ଃϔᗻᅮ⧚ᴵӊ.䖯㗠೼xoyᑇ䴶Ϟ䆕߱ؐ㾷ᄬ೼Ϩଃϔ.(2)಴Ўf(x,y)=x+sinyঞf¢(x,y)=cosy೼ᭈϾxoyᑇ䴶Ϟ䖲㓁,᠔yҹ೼ᭈϾxoyᑇ䴶Ϟ⒵䎇ᄬ೼ଃϔᗻᅮ⧚ᴵӊ.䖯㗠೼xoyᑇ䴶Ϟ䆕߱ؐ㾷ᄬ೼Ϩଃϔ.1-(3)಴Ўᮍ⿟েッߑ᭄f(x,y)=x3೼䰸এy䕈໪ⱘᭈϾxoyᑇ䴶Ϟ䖲㓁Ϩf¢(x,y)=0,᠔ҹ೼䰸এy䕈໪ⱘᭈϾxoyᑇ䴶Ϟ߱ؐ㾷ᄬ೼Ϩଃϔ.yy,y³0,(4)಴Ўᮍ⿟েッߑ᭄f(x,y)=y=೼ᭈϾxoyᑇ
-y,y<01,y>0,2y䴶Ϟ䖲㓁,㗠fy¢(x,y)=೼䰸এx䕈໪ⱘᭈϾxoyᑇ䴶Ϟ-1,y<0
2-y䖲㓁,᠔ҹ೼䰸এx䕈໪ⱘᭈϾxoyᑇ䴶Ϟ߱ؐ㾷ᄬ೼Ϩଃϔ.1dy33.䅼䆎ᮍ⿟=y3೼ᗢМḋⱘऎඳЁ⒵䎇ᅮ⧚2.2ⱘᴵӊ.ᑊ∖䗮䖛dx2(0,0)ⱘϔߛ㾷.2¶f1-㾷˖েッߑ᭄ᇍyⱘأᇐ᭄=y3,ᰒ✊ᅗ೼ӏԩϔϾϡࣙ৿x䕈¶y2(y=0)Ϟⱘ⚍ⱘ᳝⬠䯁ऎඳЁᰃ᳝⬠ⱘ,಴ℸ೼䖭⾡ऎඳЁ㾷ᰃᄬ೼ଃϔⱘ.े,া᳝䗮䖛y=0Ϟⱘ⚍ৃ㛑ߎ⦄໮Ͼ㾷ⱘᚙމ(ᮍ⿟েッⱘ䖲㓁ᗻ䆕೼ӏԩ᳝⬠ऎඳЁ,㾷ᰃᄬ೼ⱘ).ॳᮍ⿟ߚ⾏ব䞣ᕫ1-3y3dy=dx2Ϟᓣϸッপ⿃ߚᕫ15 2333y3=x-C2223y=±(x-C)2݊Ё(x-C)³0.ℸ໪᳝⡍㾷y=0.಴ℸ䖛⚍(0,0)᳝᮴か໮Ͼ㾷(བ೒(9)᠔⼎).y=0,0,x£Cy=3
(x-C)2,x>C0,x£Cy=3
-(x-C)2,x>C.೒(9)dy24.䆩⫼䗤⃵䘐䖥⊩∖ᮍ⿟=x-y⒵䎇߱ؐᴵӊy(0)=0ⱘ䖥Ԑ㾷:dxj(x),j(x),j(x),j(x)0123㾷˖j(x)=y(0)=00x12j(x)=0+(s-0)ds=x102x1221215j(x)=0+[s-(s)]ds=x-x202220x12152121518111j(x)=0+[s-(s-s)]ds=x-x+x-x.302202201604400dy225.䆩⫼䗤⃵䘐䖥⊩∖ᮍ⿟=y-x⒵䎇߱ؐᴵӊy(0)=1ⱘ䖥Ԑ㾷:dxj(x),j(x),j(x)012㾷˖j(x)=y(0)=10x13j(x)=0+(1-s)ds=1+x-x103x13222142517j(x)=1+[1+s-s)-s]ds=1+x+x-x-x+x.203615636.䆩䆕ᯢᅮ⧚2.2Ёⱘn⃵䖥Ԑ㾷j(x)Ϣ㊒⹂㾷j(x)᳝བϟⱘ䇃ᏂԄ䅵n16 ᓣ:nMNn+1j(x)-j(x)£x-xn0(n+1)!x䆕˖⬅j(x)=y+f(s,j(s))dsঞ䗁ҷ߫0x0j(x)=y,00xj(x)=y+f(s,j(s))dsn=1,2,n0xn-10ᕫxj(x)-j(x)£f(s,j(s))ds£Mx-x00x0䆒nMNn+1j(x)-j(x)£x-xn0(n+1)!߭zj(x)-j(x)£f(s,j(s))-f(s,j(s))dsn+1nx0n+1MNxn+1£s-xds0(n+1)!x0n+1MNn+2£x-x0(n+2)!nMNn+1⬅ᔦ㒇⊩ৃⶹ,ᇍӏᛣn⃵䖥Ԑ㾷,Ԅ䅵ᓣj(x)-j(x)£x-xn0(n+1)!៤ゟ.7.߽⫼Ϟ䴶ⱘԄ䅵ᓣ,Ԅ䅵:1(1)4乬Ёⱘϝ⃵䖥Ԑj(x)೼x=੠x=1ᯊⱘ䇃Ꮒ;321(2)5乬ЁⱘѠ⃵䖥Ԑj(x)೼x=ᯊⱘ䇃Ꮒ.24dy2㾷˖(1)ᰒ✊߱ؐ䯂乬=x-y,y(0)=0೼ऎඳR:x£1,y£1Ϟᄬdx೼ଃϔ㾷,⬅㾷ⱘᄬ೼ଃϔᗻᅮ⧚ⶹ,㾷ⱘᅮНऎ䯈Ўx£h0b21݊Ёh=min(a,),M=maxx-y=2.䖭䞠a=1,b=1,Ң㗠h=,े00M(x,y)ÎR217 1ᕫ㾷ⱘᅮНऎ䯈Ўx£.2߭⬅䇃ᏂԄ䅵݀ᓣnMNn+1y(x)-y(x)£x-xn0(n+1)!¶f݊ЁNᰃᴢ᱂Ꮰݍᐌ᭄.಴Ў=-2y£2,ৃপN=2,¶y1ᔧx=ᯊ,᳝232×2141y(x)-y(x)£()=.34!224ᔧx=1ᯊ,᳝32×242y(x)-y(x)£(1)=.34!3dy22(2)ᰒ✊߱ؐ䯂乬=y-x,y(0)=1೼ऎඳR:x£1,y-1£1Ϟᄬdx೼ଃϔ㾷,⬅㾷ⱘᄬ೼ଃϔᗻᅮ⧚ⶹ,㾷ⱘᅮНऎ䯈Ў:x£h0b221݊Ёh=min(a,),M=maxy-x=4.䖭䞠a=1,b=1,Ң㗠h=,े00M(x,y)ÎR41ᕫ㾷ⱘᅮНऎ䯈Ўx£.4߭⬅䇃ᏂԄ䅵݀ᓣnMNn+1y(x)-y(x)£x-xn0(n+1)!¶f݊ЁNᰃᴢ᱂Ꮰݍᐌ᭄.಴Ў=2y£2,ৃপN=2,᳝߭¶y24×2131y(x)-y(x)£()=.23!4248.೼ᴵᔶऎඳa£x£b,y<+¥؛,ݙ䆒ᮍ⿟(2.1)ⱘ᠔᳝㾷䛑ଃϔ,ᇍ݊ЁӏᛣϸϾ㾷y(x),y(x),བᵰ᳝y(x)