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'၂ᅣ༝ี၂ࢫ༝ี1.၂۱ᄅ౯ఖၛ1000/ൈषཟᄅ౯і૫Լᆰሢbູਔିᄝᄅ౯і૫ղೈሢđࠧሢൈఖᆞູਬđླေಗ၂۱ିิ܂ࡆູ-20000/ൈ2ࡨఖđ൫ษંᆃ۱ࡨఖᄝޅۚൈಗູĤࢳğഡԚđ2h/moooo2°=aູࡆđhູູۚv=1000m/h,ࢆઋ෮Ⴈൈࡗ൞TđᄵႵ08>>dh=v+at.>>>:dh|=0dtt=TႮ၂ཛॖh=vt+1at2+C,ఃᇏCູၩӈඔđႮཛॖvT+1aT2+C=0,Ⴎཛॖ0202T=°v0đջೆ۲ཛඔᆴđቋᇔh(0)=C=25m.a2.၂ݣ҂ଽѻೆੀđൈ/ٚ1৫VູਈඣAಙݣଽѻೆஆđٚ৫Vູਈඣѻ۱ಙAඣਈູV2৫ٚ/ൈđੀԛѻඣਈູV1+V2৫ٚ/ൈb2000୍ѻᇏಙA୩ູ5m0đӑݖਔݓࡅܿקѓሙđູਔᇍಙđՖ2000୍ఏཋקஆೆѻᇏඣݣಙA୩҂ӑݖm0b൫ษંѻᇏಙA୩э߄Ĥ5ࢳğഡಙA୩ູP(t),Ⴎีၩॖ8>dx=v>>dt=v1>:x(0)=0,y(0)=04.Ϝᇗ200kgุࠒູ4º౯ุބᇗ150kgุࠒູºჵᇸุൈ٢ފđԚູਬđඣቔႨᄝ༯Ӧ3౯ุބჵᇸุഈቅ৯ٳљູ∏∫cބ∏∫sđఃᇏ∫cބ∫sٳљ൞౯ุބჵᇸุđ∏൞၂۱ᆞӈඔđ൫ಒקଧ၂۱ุ༵ղඣb1
ࢳğഡਆ۱ุᇉਈٳљູmc,msđุࠒູVc,Vs,ᄵႵğ8>°(x°C1),°1<41pdy(6)y=>>0,C1:2(x°C1)4,C21pp:y(°1°p2)+a[arcsinp°p1°p2]=C|p|<1ఃᇏCູၩӈඔđႮp-љمॖᆩٚӱఅࢳĠxpx2px0(2)ჰٚӱ߄ູy=2°8p2đਆшؓxѩކѩোཛॖ(2°16p2)(1°xpp)=0đၹՎٚӱࢳ°1422342ູy=°3£23x3ࠇ2Cyx=Cx°16xđఃᇏCູၩӈඔđႮC-љمॖᆩหࢳ൞అࢳĠ(3)ჰٚӱ߄ູy=2xp+x2p4đਆшؓxѩކѩোཛॖ(p+2xp0)(1+2xp3)=0đၹՎٚӱࢳ°42422ູy=°3£23x3ࠇ(y°C)=4C|x|đఃᇏCູၩӈඔđႮC-љمॖᆩหࢳ൞అࢳĠ(4)u=siny,v=sinxđᄵ(u0)2+u0vv0°u(v0)2=0đp=du,t=dpđᄵ(2p+v)t=0đՖطdvdvp=°vࠇt=0đႮభᆀॖหࢳ4siny+sin2x=0đႮުᆀ๙ࢳsiny=Csinx+C2đఃᇏC൞2ၩӈඔđႮC-љمॖᆩหࢳ൞అࢳĠ(5)ਆшӰၛ2yđu=y2,p=u0đᄵ2u=xp°2pđਆшؓxѩކѩোཛॖ[(p+2x)2°px+2p8](xp0°p)=0đၹՎ(p+2x)2=8ࠇxp0=pđႮభᆀหࢳp=°2x±22đႮުᆀ๙pࢳp=C|x|đఃᇏC൞ၩӈඔđp=°2x±22ൈđu<0đაu=y2∏0؛đၹՎٚӱᆺႵ๙ࢳ(±Cx|x|°2y2)(±C|x|+x)=±2C|x|đఃᇏC൞ၩӈඔĠ2(6)ਆшӰၛy2đu=y2,p=u0đᄵ36p2+(8u°1)2=1đx=s,6p=sint,8u°1=costđႮႿdu=pdxđॖsint(ds+dt)=0đՖطsint=0ࠇds+dt0đႮభᆀp=0đႮުᆀs+3t=Cđ68684ᄵٚӱหࢳູy=0ࠇy=±1đ๙ࢳູsin2(4x+C)+64y4°16y2=0đఃᇏC൞ၩӈඔđႮC-љ23مॖᆩหࢳ൞అࢳĠ2.ᄝxđy૫ഈಒק౷ཌy=y(x)đऎႵᆃဢྟᇉğᄝ౷ཌy=y(x)ഈ၂(x,y)ԩ్ཌაቕѓჰOᆃ(x,y)৵ཌཌྷԼᆰbࢳğႮีၩॖᆩdyy=°1đࢳٚӱx2+y2=CđఃᇏCູၩ٤ڵӈඔbdxx3.߂ԛၛxᇠູᇠၛჰູࢊཌቂy2=4c(x+c)đᆃ۱ཌቂ෮ડቀັٳٚӱđѩᆣૼğఃᇏdyߐӮ°dxൈᆃ۱ັٳٚӱ൞҂эbࢳğਆшؓࢳіղൔܱႿxॖC=1yy0đdxdy2ᄜս߭ࢳіղൔॖັٳٚӱູy=2xy0+y(y0)2bᆣૼĠࡼdyߐӮ°dx=°1սೆٚӱࠧॖགྷٚӱ҂эbdxdyy04bೂݔ၂౷ཌቂᇏૄ၂౷ཌ۵ਸ਼၂౷ཌቂᇏૄ၂౷ཌᆞࢌ(ࠧཌྷԼᆰ)đᄵૌӫᆃਆቂ౷ཌູᆞࢌđ၂ቂ౷ཌӫູ൞ਸ਼၂ቂ౷ཌᆞࢌ݅ཌbၘࣜ۳ק၂౷ཌቂٚӱđ൫ԛਸ਼၂ቂ౷ཌٚӱđѩሹࢲ၂༯ٚمb11
ࢳğ҂ٞഡਆቂ౷ཌູ8:ax=®3(1+p2)2Ⴎުᆀ๙ࢳy=Cx+paCđఃᇏCູ҂ູਬၩӈඔbC2+16.(i)ഡሔѽሰ໙ีᇏa0đթᄝᆞᆜඔK=K(")đn>Kൈđᄝࡗxi°1∑x∑xiഈ၂ᇁ"|f(xi°1,yi°1)°f(x,"n(x))|∑Æ17
ᆃဢđn>KൈZx1XNZxif(x,"(x))+f(x,"(x))i°2i°2i°1i°1|±n(x)|∑|f(x0,y0)°f(x,"n(x))|dx+|°f(x,"n(x))|dxx0i=2xi°12Zxf(xN°2,"(xN°2))+f(xN°1,"(xN°1))+|°f(x,"n(x))|dxx2NZxi""∑N+∑"xi°1ÆnӮ৫bࢤ༯টᆣૼაज़Чᇏࠫެ၂ဢb2.০ႨAscoliႄᆣૼ༯૫ࢲંğഡ၂ݦඔਙᄝႵཋࡗIഈ൞၂ᇁႵࢸބ৵࿃đᄵᄝIഈᇀഒႵ၂۱၂ᇁ൬৻ሰਙbѩई২ඪૼđI൞ཋࡗൈഈ૫ࢲં҂၂קӮ৫bᆣૼğ҂ٞഡI=[a,b)bႮAscoliႄ่ࡱᆩđᆺྶᆣૼݦඔਙᇏ၂۱f(x)="n(x)ॖຉӮ[a,b]ഈ৵࿃ݦඔđࠧࠞཋlimf(x)թᄝࠧॖbx!b°౼၂ਙඔxn,n=1,2,...,ؓၩnđႵa∑xn0);dx8dy<0,y=0,(b)=dx:yln|y|,y6=0.ᆣૼğ(a)(b)ၹູy1°y2|y1ln|y1|°|y2ln|y2||=|y1ln|y1|°y1ln|y1°y2|+y1ln|y1°y2|°y2ln|y1°y2|+y2ln|y1°y2|°y1ln|y2||∑|y1||ln|+|y1y121
F(r)=2r+lnrᄵॖᆣb12.(a)ഡf(x),g(x),y(x)൞x0∑x∑x1ഈ٤ڵ৵࿃ݦඔ.ᆣZxy(x)∑g(x)+f(ø)y(ø)dø(x0∑x∑x1)x0ᄵZxRxf(s)dsy(x)∑g(x)+f(ø)g(ø)eødø(x0∑x∑x1)x0(b)ᄝ(a)ࡌഡ༯,g(x)ߎ൞ֆט༯ࢆ,ᄵRxf(ø)døy(x)∑g(x)ex0Rxᆣૼ:(a)ഡu(x)=f(s)x(s)dsᄵႵx0u0(x)=f(x)y(x),u0(x)°fu=f(y°u)=)u0(x)°fu∑fg=)u0(x)∑fg+fuRx°f(s)dsਆшӰၛex0RRRxxx0°f(s)ds°f(s)ds°f(s)dsu(x)ex0∑fgex0+fuex0ܱႿxՖx0xࠒٳRZxRxs°f(s)ds°f(ª)dªu(x)ex0∑f(s)g(s)ex0dsx0ZxRs°f(ª)dª=)u(x)∑f(s)g(s)ex0dsx0Ⴛၹູu(x)>y(x)°g(x)ZxRs°f(ª)dª=)y(x)∑g(x)+f(s)g(s)ex0dsx0RRx°sf(ª)dª(b)y(x)∑g(x)+f(s)g(s)ex0dsx0RRx°sf(ª)dª∑g(x)+g(x)f(s)ex0dsx0RRx°sf(ª)dª=g(x)(1+f(s)ex0ds)x0RxRtf(ª)dª0=g(x)(1°(es)sds)x0Rxf(ª)dª0=g(x)es)sds22
3.31.(a)൫ಒקൡx0ᆴ,ٚӱ12xn+1=xn°(xn°2)4pםՑᇯၬקսxn൬৻Ⴟ2.p(b)࿊౼x0=1.4.൫ᆣૼ:ູਔ211໊Ⴕིཬඔ,ေ30Ցםս.10x0ᆣૼ:(a)ഡf(x)=x°(x2°2),ᄵf(x)=1°,ေ|f(x)|∑∏<1,ࣇླ00,෮ၛႮࢺᆴקթᄝື၂x=¥,x=cosx.(b)00րק.M=sup|f(x,y)|,Æ=min{a,b}.(x,y)2RM(1);(x,y)6=(0,0);ࡌק(x,y)҂ᄝy=0ቕѓᇠഈ;a∑|x|,ᄵb<|y|,M=1,Æ=1,ᄵՎൈࢳ000000(|y0|°b)2թᄝࡗູ(°1,1);ࡌקy=0,ᄵx6=0,a<|x|,b>0;ՖطÆ=min{a,b},ᄵՎൈࢳթᄝ000(|x0|°a)2ࡗູ(2x0,0)ࠇ(0,2x0).p(2);aॖ౼ൡਈնy<0;M=(y°b)(y°b°1),Վൈ౼b=y2°y,Æ=p1,ᄵՎൈࢳ0000022y°y0+1°2y00թᄝࡗູ(x°p1,x+p1);02022y°y0+1°2y02y°y0+1°2y000(3)aॖ౼ൡਈն,Æ=min{a,b},౼bቀܔն,ᄵॖ౼Æ=1,ᄵՎൈࢳթᄝࡗູ(x°1,x+1).(|y0|+b0p0bp1+y2+y0(4);aॖ౼ൡਈն,y<0;M=1+(y°b)2,Æ=,Ֆط౼b=1+y2;Æ=0;ᄵ00b2°2y0b+y2+102pp01+y2+y01+y2+y0bՎൈࢳթᄝࡗູ(x°0,x+0)y∏0;M=1+(y+b)2,Æ=,Ֆ020200b2+2y0b+y2+1ppp0p1+y2°y01+y2°y01+y2°y0ط౼b=1+y2;Æ=0;ᄵՎൈࢳթᄝࡗູ(x°0,x+0)0202022.ഡᄝთ{(x,y):x0∑x<1,°10}ଽđીႵഥGшࢸ,ᆃ၂൞ڎაഈඍഥקཌྷ؛?൫ඪૼႮ.ᆣૼ:҂؛.ັٚٳັো)y,x(f=ydؓᆌ൞קഥط.x°=ydႿࡎ҂0=ydy+xdxӱٚٳӱ.dxydx4.൫ᆣૼğٚӱdy=y2(1°y)3eyၛ(x,y)ູԚᆴႷྛࢳቋնթᄝࡗູ[x,+1)đఃᇏx∏0dx0000đy0ၩ۳קbčีଢቔਔูߐĎᆣૼğၞᆩݦඔf(y)=y2(1°y)3eyࠣఃඔᄝᆜ۱(x,y)૫ഈ৵࿃đՖطCauchy໙ีࢳ൞ື၂bཁಖđy¥0,y¥1൞ٚӱਆ۱หࢳb(x0,y0)ᄝy=0༯ٚbࢳ౷ཌོੱնႿਬđᄵࢳ౷ཌֆט־ᄹđႻ҂ିᄀݖy=0đՖطႵࢸđႮᅚקॖᆩđႷྛࢳቋնթᄝࡗູ[x0,+1)bః౦ྙॖোරٳ༅b5.ഡԚᆴ໙ีdy2(x+y)2(E):=(y°2y°3)e,y(x0)=y0dxࢳቋնթᄝࡗູa0,ᆃඪૼy=y(x)൞ֆט־ᄹ,Ⴛ҂ିᄀݖy=°1,ᄵႮק3.5.1,b=1.dx(2)(x0,y0)ᄝy=°1აy=3ᆭࡗࠇy=°1ࠇy=3ഈ.Ⴎק3.5.1,ॖᆩa=°1,b=1.(3)(x0,y0)ᄝy=3ഈٚ.(1)ॖᆩ,a=°1.6.ࡌഡa(x)ބb(x)ᄝࡗIഈ৵࿃đᆣૼཌྟັٳٚӱdy=a(x)y+b(x),(x2I)dxૄ၂۱ࢳy=y(x)(ቋն)թᄝࡗູIbᆣૼ:ഡࡗ(a1,b1)ડቀx02(a1,b1)Ω[Æ,Ø]ΩI,ᄵݦඔf(x,y)=a(x)y+b(x)ᄝთ{(x,y)|a10,g(y)>0.ZxH(x)=h(ø)dø0ഡؓႿޅª,ࠒٳZ1døG(ª)=ªg(ø)ޚթᄝ.ᆣğ(1)ೂݔG(y0)>H(a),ᄵy(x)ᄝ0∑x∑aഈႵקၬ;(2)ೂݔG(y)∑H(a),ᄵy(x)ᄝ0∑xH(a)đᄵႵG(y(a))>0đࠧG(y(x))ᄝ0∑x∑aഈႵקၬđ෮ၛy(x)ᄝ0∑x∑aഈႵקၬb(2).ೂݔG(y0)∑H(a)đᄵႵG(y(a))∑0đႻG(ª)ޚᆞđ෮ၛy(a)҂թᄝđᄵy=y(x)сᄝ0∑x∑b0,ؓၩn,թᄝ(ª,y(ª)),|(ª,y(ª))°(x,y)|∑1,ၛ(ª,y(ª))ູԚᆴ0nnnnnn00nnnnࢳ౷ཌy=¡(x;ªn,yn(ªn))сಖ߶Ֆy=√(x)±≤0шࢸԬԛ.ᄵૄ۱Ⴗྛࢳࠇቐྛࢳ౷ཌ൮۱ԬԛсಖႵಒࢸ,ᄵᄝၛಒࢸᆞ༯(ࠇഈ)ٚЇݣᄝ√(x)°≤00@¥ؓ၂్xބ¥Ӯ৫bᆣૼğႮႿZxZxy(x,¥)=y(0)+sin(sy)ds=¥+sin(sy)ds00ᄵZx@y@y(x,¥)=1+scos(sy)ds@¥0@¥27
@yz(x,¥)=(x,¥)đᄵഈൔ߄ູ@¥Zxz(x,¥)=1+scos(sy)zds0ᄵ@z(x,¥)=xcos(xy)z,z(0)=1@x°Rx¢ၹՎz(x,¥)=expscos(sy)ds>0đ0ࠧؓ၂్xބ¥Ⴕ@y(x,¥)>0@¥3.71.2.čĎ3.ؓnࢨཌྟັٳٚӱቆԚᆴ໙ีđ൫ྻඍѩᆣૼࢳթᄝބື၂ྟקbྻඍğթᄝື၂ྟקğཌྟັٳٚӱቆdX=A(t)X+B(t)dtᄝࡗa∑t∑bഈႵࣇႵ၂۱ડቀԚᆴ่ࡱX(t0)=X0ࢳX=X(t)đఃᇏt2[a,b],X2Rn,X(t)=(x(t),···,x(t))T,t2[a,b]൞nົཟਈݦඔđၛ001nࠣA(t),B(t)ٳљ൞۳קn£nൌइᆔބnົൌཟਈݦඔđܱႿt൞৵࿃bPnPnᆣૼğקၬnົཟਈX(t)ଆູ||X(t)||=|xi(t)|đAଆູ||A(t)||=|aij(t)|bཌྟັٳٚӱቆi=1i,j=1Ԛ໙ีࡎႿࠒٳٚӱZtX(t)=X0+(A(s)X(s)+B(s))dst0ႨᇯՑЯ࣍ٚٳࠒ۱ᆃࢳمӱđࠧZtXn(t)=X0+(A(s)Xn°1(s)+B(s))dsn=1,2,···t0ၹՎđXn°1(t)൞৵࿃đᄵXn(t)္൞৵࿃đՖط၂Я࣍ཟਈݦඔਙ{Xn°1(t)}đA=supA(t)đᄵt2[a,b]Zt||Xn(t)°Xn°1(t)||∑||A(s)||||Xn°1(s)°Xn°2(s)||dst0Ⴎ݂ବمࠧ(A(b°a))n||Xn(t)°Xn°1(t)||∑n!28
n!1ൈđXn(t)οଆ൬৻ႿX(t)đѩX(t)ܱႿt2[a,b]၂ᇁ৵࿃đX(t)൞ཌྟັٳٚӱቆࢳb༯ᆣື၂ྟđഡX(t),Y(t)൞Ԛ໙ีࢳđᄵZtX(t)°Y(t)=[A(s)(X(s)°Y(s))]dst0ႮႿX(t)൞৵࿃đႮ݂ବمॖ[A(t°t)]n0||X(t)°Y(t)||∑sup||X(t)°Y(t)||t2[t0,t]n!n!1đૌॖX(t)=Y(t)b4.ӧඍѩབྷ༥ᆣૼັٳٚӱቆCauchyקbCauchyקğॉ੮ັٳٚӱቆԚᆴ໙ีdykk=fk(x,y1,···,yn),yk(x0)=y0(k=1,2,···,n)dxࡌഡႷݦඔfk,(k=1,2,···,n)ᄝთ|x°x|∑Æ,|y°yk|∑Ø0k0ଽॖၛᅚӮ൬৻ૢࠩඔđᄵԚᆴ໙ีᄝx0ਵთ|x°x0|<Ωଽթᄝ၂ቆື၂ࢳ༅ࢳyk=yk(x)đఃᇏΩ=bᆣૼğ5.ഡԚᆴ໙ี(E):y00+p(x)y0+q(x)y=0,y(x)=y,y0(x)=y00000ఃᇏp(x)ބq(x)ᄝࡗ|x°x0|>>Æ(x);f§(x,y)=ºxcos,0∑x∑1,y=0;>>x>:°x,0∑x∑1,y<°Æ(x)ಖުॉ੮Ԛᆴ໙ี§dy§(E):=f(x,y),y(0)=0.dxૌϜࡗ0∑x∑1مٚᇏ؍1.2ࢫЧٟᄜđٺnӮٳॖၛ၂่ঘᅼཌy="§(x),(0∑x∑n1)b൫ᆣૼğ8<"§(x)∏Æ(x),2∑x∑1;ೂݔnູඔ;nn:§2"n(x)∑°Æ(x),n∑x∑1;ೂݔnູఅඔ.33
ၹՎđᆃঘਙy="§(x)n7!1ൈ൞҂൬৻,Ֆط(E§)ࢳ൞҂ື၂bnᆣૼğࡼ[0,1]ࡗnٳđᄵԚᆴ໙ี(E§)ঘᅼཌіൕູXn"(x)§=f§(x,y)(x°x)+f§(x,y)(x°x)ni°1i°1ii°1nnni=1x2[0,1]ൈđy(x)=0;n1x2[1,2]ൈđy(x)=1cosº(x°1)nn2nn(1)nູඔൈđؓx2[k°1,k](k∏3)Ⴎ݂ବॖᆩğy>Æ(x)bᄵ"(x)§∏Æ(x)bnnkn(2)োර(1)ॖཌྷႋࢲંb5.ഡԚᆴ໙ีdy22(E):=(x°y)f(x,y),y(x0)=y0dxఃᇏݦඔf(x,y)ᄝಆ૫৵࿃ડቀyf(x,y)>0,y6=0ൈ.ᄵؓޅ(x0,y0),x0<0,|y0|ൡཬൈ,(E)ࢳᄝ°1a>0,x2[x,X),ᄵy(x)>0,y=y(x),x2[x,X)ֆ1011111טֆ)X,1x]2x,)x(y=y,0>)1x(y;؛1!|)xn(y|,0°X!xnაᆃ,ࢆ־ט־ശ,ᆃაxn!X°0,|y(xn)|!1؛.ՖطႷྛࢳቋնթᄝࡗູ[x0,1);,ቐྛࢳቋնթᄝࡗູ(°1,x0].6.ഡ৵࿃ݦඔf(x,y)ؓy൞־ࡨ,ᄵԚᆴ໙ีdy=f(x,y),y(x0)=y0dxႷҧࢳ൞ື၂.(ቐҧࢳ൞ڎື၂?ିई၂۱ّ২ગ?)čีଢྩᆞĎᆣૼ:୶קЌᆣਔࢳթᄝྟ.ഡႵਆ۱ࢳy(x)ބy1(x),y(x0)=y(x1),թᄝx1>x0,y(x1)>y1(x1).ª=supx0∑x0.Ⴎࠒٳᇏᆴקթᄝx2(ª,x),ડቀr0(x)=r(x1)>0.212x1°ªਸ਼၂ٚ૫,ؓႿၩx2(ª,x1)0dydy1r(x)===f(x,y)°f(x,y1)∑0(r=y°y1∏0)dxdx34
ᆃაթᄝxડቀr0(x)>0؛.22ቐҧࢳໃсື၂.২ೂdy=x2(¬x2°y),y(0)=0.dxx<07.@"8.ᆣૼz=(x,∏)ડቀཌྟັٳٚӱ@∏dz=A(x,∏)z+B(x,∏)dxބԚᆴ่ࡱz(x0,∏)=0ఃᇏ@f@fA(x,∏)=(x,"(x,∏),∏),B(x,∏)=(x,"(x,∏),∏).@y@∏(ᇿၩğyބ∏ٳљູnົਙཟਈބmົਙཟਈൈđA(x,∏),B(x,∏)ބz(x,∏)ٳљູn£nइᆔđn£mइᆔބn£mइᆔb)ᆣૼğഡԚᆴ໙ีdy=f(x,y,∏),y(x0)=y0dxࢳູy="(x,∏)đᄵZx"(x,∏)=y0+f(x,y,∏)dxx0ՖطZx@"@f@"@f=(+)dx@∏x@y@∏@∏0Zxµ∂@"=A(x,∏)+B(x,∏)dxx0@∏@"z=đཁಖ@∏@z=A(x,∏)z+B(x,∏)@xѩz(x0,∏)=0b9.൫ई২ඪૼđೂݔັٳٚӱ҂ડቀࢳື၂ྟ่ࡱđᄵࠒٳ౷ཌቂᄝअٓຶଽ္҂ି൪ቔྛᆰཌቂbdy33x2২ğ=y2,y2҂ડቀOsgood่ࡱđy=(+c),y¥0൞ٚӱࢳđࠒٳ౷ཌᄝჰڸ࣍҂ିुӮdx3ྛᆰཌቂb10.ഡG൞(x,x)૫ഈଖთđطf(x,x),f(x,x)ؓx,x৵࿃ડቀLipschitz่ࡱđᆣ(x0,x0)൞121122121212ٚӱቆdx1dx2=f1(x1,x2),=f2(x1,x2)dtdt35
అ(ࠧf(x0,x0)=f(x0,x0)=0)ԉေ่ࡱᄝႿ(x0,x0)ၩਵთଽႵൈࡗӉູၩն݅11221212؍°,ᄝ݅؍°ഈႵdx1dx2||+||∏M,(x1(t),x2(t))2°dtdtđᆃϜtुቔൈࡗđࢳx1(t),x2(t)ुቔ(x1,x2)૫ഈ໊ᇂቕѓđ෮໌݅ࣼ൞Վᄝ(x1,x2)૫ഈ݅ࠖbčีଢႵ໙ีĎ11.ᆣૼٚӱd2ydyx+=sinydx2dx၂Ўބࢳᄝ0>dx1=a(t)x+a(t)x+a(t)x,>>dt211222233>:dx3=a(t)x+a(t)x+a(t)x+tdt311322333ؓႋఊՑཌྟັٳٚӱቆࠎЧࢳቆູ(1,°1,°1)T,et(1,1+t,t)T,et(0,1,1)T36
൫ཌྟັٳٚӱቆ๙ࢳࠣડቀԚᆴ่ࡱx1(0)=x2(0)=x3(0)=0หࢳb011et0BCBCࢳğఊՑັٳٚӱቆ๙ࢳູX(t)=©(t)Cđط©(t)=B°1et(1+t)etC@A°1tetetᄵX(t)=(C+Cet,°C+C(1+t)et+Cet,°C+Ctet+Cet)T1212310231011°110RBCBCطหࢳX§(t)=©(t)t©°1(s)B(s)dsđఃᇏ©°1(t)=BB°t°tCCB(t)=BBCCt00e°e0@A@Ae°t°1+t2+ttetet012t+t+1B2CBt2C§ၹՎC=B°°2t°3C෮ၛ๙ࢳູX˜(t)=©(t)C+X(t)@2A2°t°3t°42012°et+t+t+1B2CBt2CX˜(0)=0ൈđX˜(t)=B(3°t)et°°2t°3Cb@2A2(4°t)et°t°3t°422.ഡX=P(t)e∏t൞ӈ༢ඔఊՑཌྟٚӱቆXndxi=aijxj(i=1,·,n)dtj=1ࢳđఃᇏ∏൞ӈඔđཟਈݦਈཟᆣbൔཛ؟kݖӑ҂ඔՑ൞ਈٳ၂۱ૄ)t(PඔݦඔቆdP(t)dkP(t)e∏tP(t),e∏t,···,e∏tdtdtk൞ఊՑཌྟٚӱቆཌྟܱࢳbjᆣૼğഡA=(a),(i,j=1,···,n)ႮႿdP(t)=(A°∏E)jP(t)đᄵijdtjjd(e∏tdP(t))djP(t)dtj=e∏t(∏E+A°∏E)(A°∏E)jP(t)=A(e∏t)dtdtjၹՎཟਈݦඔቆdP(t)dkP(t)e∏tP(t),e∏t,···,e∏tdtdtk൞ఊՑཌྟٚӱቆࢳđ༯ᆣఃཌྟܱđkഡCe∏tP(t)+Ce∏tdP(t)+···+Ce∏tdP(t)=0đႮႿe∏t>0đ12dtk+1dtkk෮ၛCP(t)+CdP(t)+···+CdP(t)=0b҂ཌྟཌྷܱ҂ٞഡC6=0đପહࣼႵPj°1(t)=12dtk+1dtkj°¢°1CP(t)+CP0(t)+···+CPk(t)đႮႿP(t)ૄ۱ٳਈ൞҂ӑݖkՑ؟ཛൔđପહPj°1(t)Cj12k+1ૄႷđ0=1°Cj=···=2C=1Cॖඔ༢ࢠбđൔཛ؟)1°j(°kݖӑ҂ඔՑ൞ਈٳ۱Ցඔᇀ37
؟k°jՑđၹՎ؛đՖطૌ൞ཌྟܱb3.ഡn£nइᆔݦඔA1(t)ބA2(t)ᄝࡗa0bႻႮႿdx1(t)=21222dtx2(t)>0đ෮ၛx1(t)൞ֆטᄹbೂݔt!±1ൈx2(t)҂ཟਬđପહx1(t)ड़ק߶ᄝଖ၂ൈख़ն39
Ⴟ1đᆃაx2+x2=1؛bၹՎt!±1ൈx(t)!0đପહx(t)!1bdx2(t)=°x(t)đtԉٳ1221dt1նൈx(t)>1đ෮ၛ|dx2(t)|>1đx(t)ड़ק߶ᄝଖ၂ൈख़ཬႿ0đᆃაx(t)ીႵਬ؛Ġ12dt222(d)ཁಖx1(t)ބx2(t)൞ᆜุթᄝđႮႿÆ൞x2ᄝᆞ϶ᇠ၂۱ਬx2(0)=1đᄵx(Æ)=0x0(Æ)=°1x(Æ)=1x0(Æ)=02211xx(t)=x(t+Æ)đᄵd3+x=0x(0)=x(Æ)=1x0(0)=x0(Æ)=0bႮࢳື၂ྟॖ31dt233131ᆩx1(t+Æ)=x2(t)đၹՎdx1(t+Æ)dx2(t)=x2(t+Æ)==°x1(t))x1(t)=°x2(t+Æ))x1(t°Æ)=°x2(t)dtdt෮ၛx1(t)=x1(t+4Æ)đਆшܱႿtॖx2(t)=x2(t+Æ)b8.ഡൌ༢ඔཌྟٚӱቆ(4.10.39)Ⴕೂ༯၂ུࠎЧࢳቆğX1(t),X¯1(t),···,Xr(t),X¯r(t),X2r+1(t),···,Xn(t)ఃᇏX¯(t)іൕX(t)܋ᣢگཟਈđX2r+1(t),···,Xn(t)൞ൌᆴb൫ᆣૼğؓཌྟٚӱቆޅൌᆴࢳX=X(t)đթᄝگӈඔC1,···,CrࠣൌӈඔC2r+1,···,CnX(t)=C1X1(t)+C¯1X¯1(t)+···+CrXr(t)+C¯rX¯r(t)+C2r+1X2r+1(t)+···+CnXn(t);(4.10.37)ّᆭđؓႿޅگӈඔC1,···,CrࠣൌӈඔC2r+1,···,CnđႮ(4.10.37)ൔ۳ԛݦඔ൞(4.10.39)ࢳbᆣૼğၞᆣႮ(4.10.37)ൔ۳ԛݦඔ൞(4.10.39)ࢳđսೆٚӱဒᆣࠧॖbഡY=Xi+X¯i,W=Xi°X¯i(i=1,···,r)(§)đၞᆣYW္൞ٚӱቆࢳđ༯૫ᆣૼi2i2iiiY1,···,Yr,W1,···,Wr,X2r+1,···,Xnཌྟܱbഡa1Y1+···+arYr+b1W1+···+brWr+c2r+1X2r+1+···+cnXn=0đࡼ(*)սೆॖa1b1a1b1arbrarbr(+)X1+(°)X¯1+···+(+)Xr+(°)X¯r+c2r+1X2r+1+···+cnXn=022i22i22i22iႮႿX1(t),X¯1(t),···,Xr(t),X¯r(t),X2r+1(t),···,Xn(t)൞ࠎЧࢳቆđၹՎૌཌྟܱđՖطa1=···=ar=b1=···=br=c2r+1=···=cn=0đ෮ၛY1,···,Yr,W1,···,Wr,X2r+1,···,XnཌྟܱđࠧY1,···,Yr,W1,···,Wr,X2r+1,···,Xn္൞ࠎЧࢳቆđପહޅൌᆴࢳॖႮY1,···,Yr,W1,···,Wr,X2r+1,···,XnіൕԛটđᄜϜ(*)սೆࠧॖૌ෮ေ40
ᆣૼࢲંb9.ഡ٤ఊՑཌྟັٳٚӱቆdX=A(t)X+f(t)dtᇏf(t)6=0đᆣૼ1۱+nႵ؟ᇀႵቆӱٚٳັྟཌՑఊ٤ཌྟܱࢳbᆣૼğഡX1(t),···,Xn+2(t)൞٤ఊՑࢳđYi=XiXn+2đᄵYi൞ఊՑٚӱࢳđᄵY1,···,Yn+1ཌྟཌྷܱbࡌഡX1(t),···,Xn+2(t)ཌྟܱđପહY1(t),···,Yn+1(t)္൞ཌྟܱđ؛!!༯૫ᆣႵn+1۱ཌྟܱࢳğഡX,···,X൞ఊՑࠎЧࢳቆđX§൞หࢳđᄵY=X+X§൞٤ఊՑٚӱࢳbCY+···+1nii11CX§=0đପહn+1CX+···+CX+(C+···+C)X§=011nn1n+1ೂݔC1+···+Cn+1=0đႮX1,···,XnॖၛCi¥0ĠೂݔC+···+C6=0đᄵX§=1(CX+···+CX)đପહX§൞ఊՑٚӱࢳđၹ1n+1C1+···+Cn+111nnՎf(t)=0đf(t)6=0đ෮ၛC1+···+Cn+1=0bބࢫ༝ี1.༯ਙٚӱ๙ࢳğ753(1)dx°8dx+16dx=0Ġdtdt5dt34(2)dx+4x=0Ġdt4432(3)dx°4dx+8dx°8dx+3x=0Ġdt4dt3dt2dt(4)dx=x+y°z,dy=y+z°x,dz=z+x°yĠdtdtdt(5)dx=°3x+48y°28z,dy=°4x+40y°22z,dz=°6x+57y°31zĠdtdtdt2d2y(6)dx°x+4y=0,+x°y=0Ġdt2dt2(7)dx=3x°y+3z,dy=2x+z,dz=x°y+2zĠdtdtdt(8)dx=3x+5y,dy=°5x+3yĠdtdt(9)dx=°5x°10y°20z,dy=5x+5y+10z,dz=2x+4y+9zĠdtdtdt32(10)dx°5dx+8dx°4x=e3tĠdt3dt2dt(11)y00°5y0+6y=(12x°7)e°xĠ(12)y(4)+2y00+y=sinx,y(0)=1,y0(0)=°2,y00(0)=3,y000(0)=0Ġ(13)y00°2y0+2y=4excosxĠ80)dttdtt(14)Ġ:11x(1)=,y(1)=°3341
80)dt1+t2dtt(15)b:4x(1)=0,y(1)=3ࢳğ(1)หᆘٚӱູ∏7°8∏5+16∏3=0หᆘ۴ູ0,0,0,2,2,°2,°2ࠎЧࢳቆູ1,t,t2,e2t,te2t,e°2t,te°2t෮ၛ๙ࢳູc+ct+ct2+ce2t+cte2t+ce°2t+cte°2t1234567(2)หᆘٚӱູ∏4+4=0หᆘ۴ູ±(1+i),±(1°i)ࠎЧࢳቆູetcost,e°tcost,etsint,e°tsint෮ၛ๙ࢳູcetcost+ce°tcost+cetsint+ce°tsint1234(3)หᆘٚӱູ∏4°4∏3+8∏2°8∏+3=0pหᆘ۴ູ1,1,1±2ippࠎЧࢳቆູet,tet,etcos2t,etsin2tpp෮ၛ๙ࢳູcet+ctet+cetcos2t+cetsin2t1234p(4)หᆘٚӱหᆘ۴ູ1,1±3iؓႋหᆘཟਈູğppppT3i°1°3i°1T°1°3i3i°1T(1,1,1),(1,,),(1,,)2222ppppppppppᄵ๙ࢳູcet(1,1,1)T+cet(cos3t,°1cos3t°3sin3t,°1cos3t+3sin3t)T+cet(°sin3t,°3cos3t+12222232pppp1sin3t,3cos3t+1sin3t)T222(5)หᆘٚӱหᆘ۴ູ1,2,3ؓႋหᆘཟਈູğ(3,2,3)T,(4,1,1)T,(2,2,3)Tᄵ๙ࢳູcet(3,2,3)T+ce2t(4,1,1)T+ce3t(2,2,3)T123p(6)หᆘٚӱหᆘ۴ູ±i,±3ؓႋหᆘཟਈູğppp13Tp13T1iT1°iT(1,3,°,°),(1,°3,°,),(1,i,,),(1,°i,,)22222222ppᄵ๙ࢳູce3t(1,°1)T+ce°3t(1,°1)T+c(cost,1cost)T+c(sint,1sint)T12223242(7)หᆘٚӱหᆘ۴ູ0,2,3ؓႋหᆘཟਈູğ(°1,3,2)T,(°1,1,0)T,(4,3,1)T42
ᄵ๙ࢳູc(°1,3,2)T+ce2t(°1,1,0)T+ce3t(4,3,1)T123(8)หᆘٚӱหᆘ۴ູ3±5iؓႋหᆘཟਈູğ(1,i)T,(1,°i)Tᄵ๙ࢳູce3t(cos5t,°sin5t)T+ce3t(cos5t,°sin5t)T12(9)หᆘٚӱหᆘ۴ູ5,2±iؓႋหᆘཟਈູğ(2,0,°1)T,(20+10i,15°5i,°14°2i)T,(20°10i,15+5i,°14+2i)Tᄵ๙ࢳູce5t(2,0,°1)T+ce(2+i)t(20+10i,15°5i,°14°2i)T+ce(2°i)t(20°10i,15+5i,°14+2i)T123(10)ఊՑٚӱหᆘٚӱູ∏3°5∏2+8∏°4=0หᆘ۴ູ1,2,2ࠎЧࢳቆູet,e2t,te2t෮ၛఊՑٚӱ๙ࢳູcet+ce2t+cte2t123ഡหࢳູAe3tđսೆٚӱॖA=12ၹՎ๙ࢳູcet+ce2t+cte2t+1e3t1232(11)ఊՑٚӱหᆘٚӱູ∏2°5∏+6=0หᆘ۴ູ2,3ࠎЧࢳቆູe2x,e3x෮ၛఊՑٚӱ๙ࢳູce2x+ce3x12ഡหࢳູAxe°xđսೆٚӱॖA=1ၹՎ๙ࢳູce2x+ce3x+xe°x12(12)ఊՑٚӱหᆘٚӱູ∏4+2∏2+1=0หᆘ۴ູi,i,°i,°iࠎЧࢳቆູcosx,xcosx,sinx,xsinx෮ၛఊՑٚӱ๙ࢳູc1cosx+c2xcosx+c3sinx+c4xsinxഡหࢳູx2(Acosx+Bsinx)đսೆٚӱॖA=°1,B=082ၹՎ๙ࢳູccosx+cxcosx+csinx+cxsinx°xcosx123482ᄜႮԚ่ࡱđᄵcosx+xcosx°3sinx+13xsinx°xcosx88(13)ఊՑٚӱหᆘٚӱູ∏2°2∏+2=0หᆘ۴ູ1±i43
ࠎЧࢳቆູexcosx,exsinx෮ၛఊՑٚӱ๙ࢳູcexcosx+cexsinx12ഡหࢳູxex(Acosx+Bsinx)đսೆٚӱॖA=0,B=2ၹՎ๙ࢳູcexcosx+cexsinx+2xexsinx12(14)Ⴎ၂۱ٚӱॖࢳx=tđࡼఃսೆ۱ٚӱđॖy=°t332(15)Ⴎ၂۱ٚӱॖࢳx=0đࡼఃսೆ۱ٚӱđॖࢳy=t+1b3t2.ഡ"(t)൞ٚӱx00+k2x=f(t)ࢳđఃᇏk൞ӈඔđݦඔf(t)ᄝࡗ[0,+1)ഈ৵࿃b൫ᆣૼğ(a)k6=0ൈđٚӱ๙ࢳॖіൕູZtc21"(t)=c1coskt+sinkt+sink(t°s)·f(s)ds;kk0(b)k=0ൈđٚӱ๙ࢳॖіൕູZtx=c1+c2t+(t°s)f(s)ds0ఃᇏc1,c2൞ၩӈඔbRtᆣૼğ(a)k6=0ൈđၞᆣ"(t)=1sink(t°s)·f(s)dsູٚӱx00+k2x=f(t)၂۱หࢳđطఊՑٚ0k0ӱx00+k2x=0๙ࢳູC2x=C1coskt+sinktkܣٚӱ๙ࢳູZtC21"(t)=C1coskt+sinkt+sink(t°s)·f(s)dskk0Rt(b)k=0ൈđၞᆩ"(t)=(t°s)f(s)dsູٚӱx00(t)=f(t)၂۱หࢳđطఊՑٚӱx00=0๙ࢳ00ູx=C1+C2tđܣٚӱ๙ࢳູZt"(t)=C1+C2t+(t°s)f(s)ds0ఃᇏC1,C22Rb3.۳קٚӱx000+5x00+ax0=f(t)đఃᇏf(t)ᄝ(0,+1)ഈ৵࿃đഡ"(t),"(t)൞ഈඍٚӱၩਆ۱ࢳđ12ࠞཋlim["1(t)°"2(t)]թᄝđ൫ҕඔaᄍྸٓຶbt!+1ࢳğy="(t)°"(t)đᄵy൞ఊՑٚӱࢳđหᆘٚӱູ∏3+5∏2+a∏=0đࢳႵ∏=0,∏=1212pp°5+25°4a°5°25°4a2,∏3=2b°5tsqrt4a°25°5tsqrt4a°2525°4a<0đ๙ࢳູy=C1+C2e2cost+C3e2sintđlimyթᄝĠ22t!+1°5t°5t25°4a=0đ๙ࢳູy=C1+C2e2+C3te2đlimyթᄝĠt!+144
25°4a>0đ๙ࢳູy=C+Ce∏2t+Ce∏3tđေlimyթᄝࣇ∏,∏<0đՎൈa>0b12323t!+1ሸഈ෮ඍđa>0b4.ഡa,b൞ӈඔđf(t)൞ࡗ(°1,+1)ഈ৵࿃ݦඔbഡ!(t;ø)൞Ԛᆴ໙ีx00+ax0+bx=0,x(0)=0,x0(0)=f(ø)ࢳđఃᇏø൞၂۱ҕඔb൫ᆣૼğZtx(t)=!(t°ø;ø)dø0൞Ԛᆴ໙ีx00+ax0+bx=f(x),x(0)=0,x0(0)=0ࢳbࢳğսೆဒᆣࠧॖb5.ࢳߞᆒሰᄝቅ༯఼௧ᆒٚӱd2xm+kx=pcos!tdt2qqఃᇏm,k,pބ!൞ᆞӈඔbؓຓࡆੱ!6=kބ!=kਆᇕ҂౦ঃđٳљඪૼఃࢳၩqmmၬđᆃkіൕߞᆒሰܥႵੱbmࢳğٚӱॖ߄ູd2xkp+x=cos!tdt2mmq!6=kൈ๙ࢳູmrrkkpx(t)=C1cost+C2sint+k2cos!tmmm(°!)m(ຓࡆཛ༢ሱಖੱ٤܋ᆒൈđ༢ᆒږႵཋ)q!=kൈđ๙ࢳູmrrrkkpkx(t)=C1cost+C2sint+ptsintmm2kmm(ຓࡆཛ༢ሱಖੱԩႿ܋ᆒሑൈđႄఏ༢ᆒږཋᄹն)6.ᆣૼğӈ༢ඔఊՑཌྟັٳٚӱቆޅࢳx!1ൈႿਬđࣇ༢ඔइᆔA෮Ⴕหᆘ۴ऎႵڵൌbᆣૼğӈ༢ඔఊՑཌྟٚӱቆޅࢳx!1ൈႿਬđࣇٚӱቆࠎЧࢳቆx!1ൈ45
Ⴟਬbૌॖၛܒᄯٚӱቆ၂۱ࠎЧࢳቆೂ༯ğY1(x),···,Yn1(x);···;Y1(x),···,Ynk(x)11kknj°1XxiYl(x)=e∏jx[(A°∏E)i]vljji!i=0ॖေམx!1ൈႿਬđหᆘ۴сྶऎႵڵൌbࢫ༝ี1.ഡᄝࡗ(°1,+1)ഈq(x)∑0đᆣૼఊՑཌྟັُ၂۱٤ޅ0=y)x(q+0y)x(p+00yӱٚٳࢳቋ؟ᆺႵ၂۱ਬb2.ᆣૼఊՑཌྟັٳٚӱy00+p(x)y0+q(x)y=0ޅਆ۱ཌྟܱࢳਬ൞ཌྷࢌհb2ᆣૼğഡy,y൞ਆ۱ཌྟܱࢳđ০Ⴈэߐy(x)=v(x)u(x)ၛࠣv>0đॖၛुu,u൞ٚӱdu+1212dt2Q(t)u=0ཌྟܱࢳđႮज़Чં4.5.2ॖ෮ေࢲંb3.ഡັٳٚӱd2x+P(t)x=0dt2ఃᇏP(t)൞t৵࿃2ºᇛ௹ݦඔđطડቀn20༯૫҂ൔӮ৫ğ(n+")22ºbၹn+"ՎࢲંӮ৫bੂࢫ༝ี1.ࢳ༯ਙшᆴ໙ีğ(1)y00+y=x;y(0)=2,y(º)=1Ġ2(2)y00°3y0+2y=ex,y(0)=0,y(1)=0Ġ46
(3)x2y00°3xy0+3y=0,y0(1)+y(1)=°2,2y0(2)°y(2)=0bࢳğ(1)ఊՑٚӱหᆘٚӱູ∏2+1=0หᆘ۴ູ±iࠎЧࢳቆູcosx,sinxఊՑٚӱ๙ࢳູc1cosx+c2sinxഡหࢳູAxđսೆٚӱA=1ၹՎ๙ࢳູc1cosx+c2sinx+xᄜ০Ⴈшᆴ่ࡱy(x)=2cosx+(1°2)sinx+xº(2)ఊՑٚӱหᆘٚӱູ∏2°3∏+2=0หᆘ۴ູ1,2ࠎЧࢳቆູex,e2xఊՑٚӱ๙ࢳູcex+ce2x12ഡหࢳູAxexđսೆٚӱA=°1ၹՎ๙ࢳູcex+ce2x°xex12ᄜ০Ⴈшᆴ่ࡱy(x)=1ex+1e2x°xex1°ee°1(3)ႮEulerٚӱࢳم๙ࢳູcx+cx312ᄜႮшᆴ่ࡱy(x)=°xb2.ᆣૼшᆴ໙ี8º(s°º)5p3º(2)F(s)=5đs>°44(s+4)2(3)F(s)=3ºđs>°2(s+2)2+9º2pp(4)F(s)=22s+52đs>°1(2s+1)2+1622.࠹ෘ༯ਙݦඔF(s)ঘ൦эߐğ(1)F(s)=3,(2)F(s)=s°12s°4(s+1)2(3)F(s)=s+2,(4)F(s)=1s2+4s+5s2°43(5)F(s)=s,(6)F(s)=5°2s(s°4)4s2+7s+103(7)F(s)=s,(8)F(s)=s(s2+k2)2s4+4a4ࢳğ(1)f(t)=3e2t2(2)f(t)=(1°2t)e°t48
(3)f(t)=e°2tcost(4)f(t)=1sh(2t)2(5)f(t)=e4t(1+12t+24t2+32t3)3°7t3°7t3(6)f(t)=°2e2ch(t)+8e2sh(t)22(7)f(t)=tsin(kt)2k(8)f(t)=1eatcos(at)+1e°atcos(at)223.Ⴈঘ൦эߐٚمࢳ༯ਙԚᆴ໙ีğ(1)y00+4y0+4y=t2;y(0)=y0(0)=0;(2)x00+6x0+34x=30sin2t;x(0)=x0(0)=0(3)x00+!2x=Fsin!t,x(0)=x0(0)=0,!6=0;000(4)y(4)+2y00+y=4tet,y(0)=y0(0)=y00(0)=y(3)(0)=0.ࢳğ(1)ᄝٚӱਆҧ౼ঘ൦эߐđѩ࠺L{f(t)}=Y(s)đ22sY+4sY+4Y=.s32෮ၛY=2đy(t)=°te°2t°3e°2t+t°t+3s3(s2+4s+4)48428(2)ᄝٚӱਆҧ౼ঘ൦эߐđѩ࠺L{x(t)}=X(s)đ260sX+6sX+34X=.s2+4෮ၛX=60đx(t)=60(e°3tcos(5t)°2e°3tsin(5t)°cos(2t)+5sin(2t))(s2+4)(s2+6s+34)17452(3)ᄝٚӱਆҧ౼ঘ൦эߐđѩ࠺L{x(t)}=X(s)đ22!sX+!0X=F0s2+!2.෮ၛ!=!ൈީđX=F!0đx(t)=F0sin(!0t)°F0tcos(!0t)b00(s2+!2)22!22!000!6=!ൈީđX=F(1°1)!đx(t)=F0(!0sin(!t)°!sin(!0t))b00s2+!2s2+!2!2°!2!0(!2°!2)000(4)ᄝٚӱਆҧ౼ঘ൦эߐđѩ࠺L{f(t)}=Y(s)đ424sY+2sY+Y=.s2෮ၛY=4đy(t)=2tcost°6sint+4tbs2(s4+2s2+1)4.Ⴕ၂ᇕԯѷࡨᆑఖ০Ⴈ൞ᯒކᆒჰbᄝඣ૫ഈԯุॖၛቔູ၂۱ቅᆒሰđఃᇉਈູMđ࣑༢ඔູKđቅ༢ඔູRđࡨᆑఖᄵ൞ڸᄝԯุഈᆒሰđᇉਈູmđ࣑༢ඔູkbԯุ໊၍X(t)ބࡨᆑఖ໊၍x(t)ડቀᄎٚӱ80)Ġ(2)(2x+1)2y00°4(2x+1)y0+8y=0Ġ2(3)dx+x=1Ġdt21+cos2t2(4)t2(t+1)dx°t(2+4t+t2)dx+(2+4t+t2)x=°t4°2t3Ġdt2dt2(5)tdx°(2t+1)dx+(t+1)x=(t2+t°1)e2tĠdt2dt32(6)(1°t2)dx°tdx+dx=0bdt3dt2dtࢳğ(1)x=etđᄵy+4y+13y=0đหᆘٚӱ∏2+4∏+13=0đ∏=°2±3iđႿ൞y=e°2t(ccos3t+ttt1c2sin3t)đ෮ၛc1cos(3lnx)+c2sin(3lnx)y=.x2(2)2x+1=t,ᄵt2y°2ty+2y=0đᄜt=euđᄵหᆘٚӱູ∏(∏°1)°2∏+2=0đࢳ∏=1,2btttႿ൞y=ceu+ce2u=ct+ct2=c(2x+1)+c(2x+1)2.121212(3)ఊՑٚӱหᆘٚӱູ∏2+1=0đႿ൞ఊՑٚӱႵ๙ࢳx(t)=ccost+csintđಖުႮӈඔэၳ܄12ൔႵZtsin(t°s)x(t)=c1cost+c2sint+ds.t1+cos2s0(4)ૌ൮༵ھٚӱหࢳđҩࢳྙൔູx=at2+bt+cđսೆٚӱa=1,c=0bႿ൞Ⴕหࢳx=t2+btbႮՎૌᆩx=t൞ཌྷႋఊՑٚӱࢳbૌᄜႨࢆࢨمটఊՑٚӱ๙ࢳđࠧഡࢳູx=tu(t)đսೆٚӱt3[(t+1)u°(t+2)u]=0đv=uđᄵࢳv=C(t+1)etđႿ൞u=Ctetđtttt෮ၛቋᇔૌٚӱ๙ࢳູx=t2+Ct+Ct2et.12(5)ૌ൮༵ھٚӱหࢳđҩࢳྙൔູx=(at+b)e2tđսೆٚӱa=1đb=0đႮՎᆩห50
ࢳູx=te2tbᄜఊՑٚӱਆ۱ཌྟܱࢳđᇿၩఊՑٚӱູ0dx2dxdx2dxdxt°(2t+1)+(t+1)x=t(°)°(t+1)(°x)=0,dt2dtdt2dtdtႿ൞ಸၞหࢳx=CetđಖުႨࢆࢨمđഡx=C(t)etđսೆٚӱđࢳx=t2etbႿ൞ቋᇔٚӱ12ࢳູx=(C+Ct2)et+te2t.122P(6)y=dxđᄵھٚӱູ(1°t2)dy°tdy+y=0đႨૢࠩඔمࢳđഡࢳູy=atnđսೆٚdtdt2dtn∏0nӱđᄵႵa=n°3abႿ൞ૌႵ๙ࢳy=ct+c(1°1t2°Pk∏2(2k°3)!!t2k)bႮՎჰটٚӱnnn°2122(2k)!!๙ࢳູ1X(2k°3)!!t2k+1x=c+ct2+c(t°t3°k∏2).0126(2k)!!2k+12.ԛ༯ਙັٳٚӱᄝx=x0ԩᅚषਆ۱ཌྟܱૢࠩඔࢳđѩཿԛཌྷႋ־܄ൔğ(1)y00°xy0°y=0,x=0;(2)y00°xy0°y=0,x=1;003(3)(1°x)y00+y=0,x=0;(4)dy+xy=0,x=0;0dx302(5)xdy+4dy+xy=0,x=0;(6)y00+(sinx)y=0,x=0.dx2dx00ࢳğPP(1)y=axnđսೆٚӱ(a(n+2)(n+1)°an°a)xn=0đႿ൞־܄n∏0nnn+2nnan°2ൔan=nb෮ၛೂݔٳљ౼(a0,a1)=(1,0)ބ(a0,a1)=(0,1)đૌॖၛਆ۱ཌྟܱࢳXx2ky1=,(2k)!!k∏0Xx2k+1y2=.(2k+1)!!k∏0PP(2)y=a(x°1)nđࡼٚӱཿູy00°(x°1)y0°y0°y=0đսೆٚӱ(a(n+2)(n+1)°n∏0nnn+2an°a(n+1)°a)(x°1)n=0đႿ൞־܄ൔa=an°1+an°2b෮ၛೂݔٳљ౼(a,a)=(1,0)nn+1nnn01ބ(a0,a1)=(0,1)đૌॖၛਆ۱ཌྟܱࢳbPP(3)y=axnđսೆٚӱ(a(n+2)(n+1)°a(n+1)n+a)xn=0đႿ൞־n∏0nnn+2n+1nan°1(n°1)(n°2)°an°2܄ൔan=n(n°1)b෮ၛೂݔٳљ౼(a0,a1)=(1,0)ބ(a0,a1)=(0,1)đૌॖၛਆ۱ཌྟܱࢳbPP(4)y=axnđսೆٚӱ6a+[a(n+3)(n+2)(n+1)+a]xn=0đႿ൞n∏0n3n∏1n+3n°151
־܄ൔa3=0,an=°an+4(n+4)(n+3)(n+2)(n∏0)b෮ၛೂݔٳљ౼(a0,a1,a2)=(1,0,0)ބ(a0,a1,a2)=(0,1,0)đૌॖၛਆ۱ཌྟܱࢳbPP(5)y=axnđսೆٚӱ4a+[a+(n+4)(n+1)a]xn=0đႿ൞־܄n∏0n1n∏1n°1n+1ൔa=0,a=°an(n∏0)b෮ၛೂݔٳљ౼a=1ބa=2đૌॖၛਆ۱ཌྟܱ1n+2(n+5)(n+2)00ࢳb3.ؓႿ༯ਙԚᆴ໙ีԛy00(x),y(3)(x)ބy(4)(x)đՖطཿԛཌྷႋԚᆴ໙ีࢳᄝxীࠩඔ0000భ5ཛğ(1)y00+xy0+y=0;y(0)=1,y0(0)=0;(2)y00+(sinx)y0+(cosx)y=0;y(0)=0,y0(0)=1.ࢳ:(1)Ⴎٚӱđૌᆩy(k+2)=°xy(k+1)°(k+1)y(k),Ⴟ൞ႮԚᆴđૌy00(0)=°1,y(3)(0)=0ބy(4)(0)=3đႮՎx2x4y=1°++···,28(2)ႮٚӱބԚᆴđૌႵy00(0)=0đؓٚӱ၂ࢨඔđy(3)(0)=°2ބy(4)(0)=0đႮՎx3y=x°+···.34.ࢳHermiteٚӱğy00+2xy0+∏y=0,(°10ൈđr1=i∏,r°2=°i∏đٚӱ๙ࢳູy=C1cos∏t+C2sin∏tđսೆшᆴ่ࡱॖ8p0༯૫҂ൔӮ৫ğ2g(x1)°g(x2)2(n+")<<(n+1),(°1>°x0+y0+z0°x+2z=e°t>>>:x0+y0°z+x+2y=3e°tࢳğ8>>x=1Ct2e°t°Cte°t+Ce°t+5te°t°t2e°t+1t3e°t><212324y=°Cte°t+Ce°t+2te°t°3t2e°t>>124>:z=Ce°t+3te°t12ၞᆩޙ(°1,0)໗ק҂ࡶ࣍໗קb6.ᆣૼшᆴ໙ี80,q>0ൈđਬࢳ൞ࡶࣉ໗קb(2)p>0,q=0ࠇp=0,q>0ൈđਬࢳ൞໗קđ҂൞ࡶࣉ໗קb(3)ᄝః౦ྙ༯đਬࢳ൞҂໗קbᆣૼğഡ∏1,∏2ູAਆหᆘ۴đᄵğ∏1+∏2=°p,∏1∏2=q(1)p>0,q>0ᄵ∏1,∏2ູਆ۴ൌڵࠇ၂ؓൌູڵ܋ᣢྴ۴đႮק5.1.1ᆩđਬࢳࡶࣉ໗קb(2)p>0,q=0ᄵ∏1,∏2ק໗ࣉࡶ٤đק໗ࢳਬđᆩ1.1.5קႮđ۴ਬ۴၂ᆞ၂ູљٳbp=0,q>0ᄵ∏1,∏2ູਆ܋ᣢՂྴק໗ࣉࡶ٤đק໗ࢳਬđᆩ1.1.5קႮđ۴b(3)(1)(2)ק໗҂ࢳਬđᆩ1.1.5קႮđᆞູൌ၂۱Ⴕഒᇀ۴ᆘหđ༯ྙ౦ఃᄝđ༅ٳb4.ษંົٚӱx0=y°xf(x,y),y0=°x°yf(x,y)ਬࢳ໗קྟđఃᇏݦඔf(x,y)ᄝ(0,0)ڸ࣍൞৵࿃ॖັbࢳğf(x,y)ᄝ(0,0)ڸ࣍৵࿃ॖັđᄵf(x,y)ᄝ(0,0)၂ཬਣთଽॖ၂ࢨ࣍රᅚषູğpf(x,y)=f(0,0)+fx(0,0)x+fy(0,0)y+o(x2+y2)ᄵჰົٚӱ၂ࢨཌྟ࣍රٚӱູx0=y°f(0,0)xy0=°x°f(0,0)y61
ᄵႮഈีٳ༅ॖᆩđf(0,0)>0ൈđਬࢳࡶࣉ໗קbf(0,0)0ൈđਬࢳ໗ק໗ࣉࡶ٤đקbf(0,0)>0ൈđਬࢳ҂໗קb5.ഡݦඔg(x)৵࿃đx6=0ൈxg(x)>0.൫ᆣٚӱx”+g(x)=0ਬࢳ൞໗קđ҂൞ࡶࣉ໗קbRxᆣૼğ໗קྟᆣૼđ߄ູٚӱቆđѩ౼V=1y2+f(s)ds20קႨ০ྟק໗࣍ࡶ٤ၬᆣૼb6.ษં༯ਙٚӱਬࢳ໗קྟğ(1)x0=°y°xy2,y0=x°x4y(2)x0=°y3°x5,y0=x3°y5(3)x0=°x+2x(x+y)2,y0=°y3+2y3(x+y)2(4)x0=2x2y+y3,y0=°xy2+2x5ࢳğ(1)122V(x,y)=(x+y)>02dV(x,y)2242=°(xy+xy)<0dtਬࢳࡶࣉ໗קb(2)V(x,y)=(x4+y4)>0dV(x,y)88=°4(x+y)<0dtਬࢳࡶࣉ໗קb(4)N={(x,y):x>0,y>0}V(x,y)=x2y462
dV(x,y)2773=2xy+8xydtdV(x,y)NଽႵV(x,y)>0,>0,шࢸഈႵV(x,y)=0,dtܣਬࢳ҂໗קb7.ษંa౼ޅᆴൈđٚӱቆ80,3ีٳ༅đਬࢳ҂໗קb(2)a=0,V(x,x)=(x2+x2)>0ᄵ1212dV(x,y)=2(x4+x4)>0,ܣਬࢳ҂໗קbdt12(3)a<0,3ีٳ༅đਬࢳࡶࣉ໗קb8.࣮ࢨັٳٚӱx0=y,y0=°1+x2ਆ۱ޙ໗קྟbࢳğၞᆩٚӱቆਆ۱ޙູ(1,0),(°1,0)b(1)ؓႿޙ(1,0)bx1=x°1,y1=yᄵཌྷॢࡗ(x,y)ᇏޙ(1,0)ؓႋႿཌྷॢࡗཌྷॢࡗ(x1,y1)ᇏޙ(0,0)bႵğ80,y1>0},V(x1,y1)=x1y1ᄵႵğdV(x1,y1)2=3x1y1+x1y1dtdV(x1,y1)ᄝNഈႵğV(x1,y1)>0,dt>0ᄝ@NഈႵğV(x1,y1)=0Ⴎק໗҂)0,1(ޙğᆩקקb(2)ؓႿޙ(°1,0)bx2=x+1,y2=yᄵཌྷॢࡗ(x,y)ᇏޙ(°1,0)ؓႋႿཌྷॢࡗཌྷॢࡗ(x2,y2)ᇏޙ(0,0)bႵğ80dµr=1,dt:0,r=0൫ቔԛჰڸ࣍ཌྷđѩ࣮ޙr=0໗קྟᇉbčĎ2.؎༯ਙٚӱఅ(0,0)োđѩቔԛھఅڸ࣍ཌྷb(1)x0=4y°x,y0=°9x+y(2)x0=2x+y+xy2,y0=x+2y+x2+y2(3)x0=2x+4y+siny,y0=x+y+ey°1(4)x0=x+2y,y0=5y°2x+x3(5)x0=x(1°y),y0=y(1°x)ࢳğ(1)ᇏྏb(2)҂໗קචཟࢲb(3)λb(4)λb(5)҂໗קྒྙࢲb3.൫ಒקٚӱቆ80đթᄝ±>0đ||x1||<±Ⴕ||x1(t)=√(t)√°1(t)x1||<"b෮ၛؓႿ000၂xđሹթᄝx1đડቀx1=Cxđ||x1||<±đପહႵ0000)||x(t)||=||C√(t)√°1(t)x1||0ބt0∏0b҂ં±>0؟હཬđሹթᄝx0đෙಖ||x0||<±đ൞(4.1)Ⴕၛx(t0)=x0ູԚᆴࢳx(t,t0,x0)ᄝtႿଖt1(∏t0)ൈႵ||x(t1,t0,x0)||∏"ପહx1=Cxđᄵ0"01°11C||x(t)=√(t)√(t0)x0||=||x(t)||>C"Ӂള؛b෮ၛਬࢳ໗קb65
(2))ؓႿၩ۳קt0∏0đթᄝ±>0đᆺေx0ડቀ||x1||<±0༢(4.1)ડቀԚᆴ่ࡱx1(t)=xࢳx1=x1(t,t,x)ьႵ0000limx1(t,t,x1)=000t!+1෮ၛؓႿ၂xđሹթᄝx1đડቀx1=Cxđ||x1||<±đପહႵ0000)limx(t,t,x)=Climx1(t,t,x1)=00000t!+1t!+1(ؓႿ၂x(t)ડቀlimt!+1x(t)=0đପહؓԚᆴޓཬx(t)္ડቀlimt!+1x(t)=0bൈႮႿlimt!+1x(t)=0đᄵ၂x(t)ႵࢸđᄵႮ(1)ᆩਬࢳ໗קb෮ၛਬࢳࡶ࣍໗קb2.ഡඔᆴݦඔf(x,t)ᄝთ{(t,x):t∏0,|x|0,x2<0ٚӱડቀԚᆴ่ࡱx(0)=x1,x(0)=x2ࢳđt!+1ൈႿਬđᄵਬࢳ൞ࡶࣉ໗קbᆣૼğഡԚᆴx1,x2ؓႋࢳٳљູ"1(t),"2(t)b(1)؎ğ00,{t}1,t!1(j!1)st|"(t)|>"bၹູlim"(t)=0đ෮ၛթᄝଖ0jj=1jj0t!11۱t0đt>t0ൈđ|"1(t)|<"0đࠧթᄝi>0đj>iൈđ|"(tj)|>|"1(tj)|đطx0x0>x2ൈđၛx0ູԚᆴࢳ"(t)t!+1ൈႿਬb౼±=min{x1,°x2}đᄵ|x0|<±ൈđၛx0ູԚᆴࢳ"(t)ડቀlimt!1"(t)=0đܣਬࢳ൞ࡶ࣍໗קb3.ॉ੮༯ਙਆ۱ٚӱቆdX={A+B(t)X}dtdX=AXdtR1ఃᇏA൞ӈइᆔđB(t)൞ᄝt∏0ഈ৵࿃इᆔݦඔડቀ|B(t)|dt<10ᆣğೂݔު၂۱ٚӱ၂్ࢳᄝt∏0ഈЌӻႵࢸđᄵభ၂۱ٚӱ၂్ࢳᄝt∏0ഈ္ЌӻႵࢸb66
ᆣૼğ၂۱ٚӱቆൡކԚᆴ่ࡱX(t0)=X0ࢳॖіൕູğZtX(t)=©(t)©°1(t)X+©(ø)©°1(ø)B(ø)X(ø)dø(t∏0)00t0ఃᇏ©(ø)൞۱ٚӱቆൡކԚᆴ่ࡱ©(t0)=Eࠎࢳइᆔbၹູ۱ٚӱ၂్ࢳᄝt∏0ഈЌӻႵࢸđ©(t)©°1(t)X൞۱ٚӱቆડቀԚ่ࡱX(t)=XࢳđܣթᄝᆞӈඔMđ0000||©(t)||||©°1(t)||||X||∑M,||©(ø)||||©°1(ø)||∑M(t∏0)00ܣॖğZt||X(t)||∑M+M||B(ø)||||X(ø)||dø(t∏0)t0ႮGrowall҂ൔđᄵႵğRtM||B(ø)||dø||X(t)||∑Met0(t∏0)R+1RtMhၹູ||B(ø)||dø<+1෮ၛ||B(ø)||døႵࢸđഡູhđᄵႵğ||X(t)||∑Me(t∏0)ࠧ၂۱ٚ0t0ӱቆ၂్ࢳ൞Ⴕࢸb4.ॉ੮༯ਙਆ۱ٚӱቆdX=AX+R(t,X)dtdX=AXdtఃᇏA൞ӈइᆔđR(t,X)ᄝ{(t,X):t∏t0,||X||||)t(X||đൈ±>||X||đ"=±ᄝթđ0>"۳bMeMh05.ഡཌྟັٳٚӱx0=ax+by,y0=cx+dyၛ(0,0)ູۚࢨఅđ൫ቔԛఃཌྷbčĎ6.ॉ੮ࢨٚӱቆ80ູӈඔĎᇀഒႵ၂۱о݅b68'
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