GiiTchC2NguynThThuVn-TrnVKhanhi hcKhoaHcTNhin3/2010NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 1/79CchtnhimmnhcKimtragiahck:30%Kimtracuik:70%NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 2/79Ti liuthamkho1.C.Khanh:Giitchhmnhiubin,NXBHQGTp.HCM(2003)2N.T.LongvN.C.Tm:TonCaoCpC1,KhoaKinhTHQGTpHCM(2004)3RaymondN.Greenwell,NathanP.Ritchey,andMargaretL.Lial:CalculusWithApplicationsForTheLifeSciences,AddisonWesley(2003)4StewartJ.:Calculus-ConceptsandContexts,Brooks-Cole(2002)MtsphnmmhtrtnhtonMaxima Mathematica Maple MatlabNguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 3/79Chng1.ViPhnHmNhiuBin1.VdmuSafedivingrequiresanunderstandingofhowtheincreasedpressurebelowthesurfaceaffectsthebodysintakeofnitrogen.Itcanbeunderstoodasfollows:Adivermustchooseasafetimeforagivendepth,orasafedepthforagiventime.Henceoneinvestigatesafunctionoftwovariables,depthanddivetime.Partialderivativestellushowthisfunctionbehaveswhenonevariableisheldconstantastheotherchanges.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 4/79Th d1:Thtchcahnhtrcxcnhbicngthc:V = r2h.Rrngthtchhnhtrphthucbnknhr vchiucaohcahnhtr.VytacthvitnhsauV(r , h) = r2hTrnghpny,Vcxemnh1hmphthucvo2binsrvhTngt,thtchca1hnhhpchnhtlV = lwh.Vytacthvitnhsau:V(l , w, h) = lwh.Trnghpny,Vcxemnh1hmphthucvo3binsl , wvhNguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 5/79Th d2:Nhittiim (x, y)trnbmtkimloiphngcchobicngthcT(x, y) =601+x2+y2 ,trongTcobng0Cvx, yobngmt.Tmtcthayinhittngngvikhongcchtiim (2, 1)theoxvtheoy?Th d3:Hmchsnhit(wind-chillindex)cmtbicngthcsau:W = 13.12 +0.6215T 11.37v0.16+0.3965Tv0.16trongTbiuthnhit(0C)vvbiuthvntcgi(km/h).KhinhitT = 150Cvv = 30km/h,hydonchsnhitsgimbaonhiununhitgim10C?vnuvntcgitng1km/h?NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 6/79Chng1.ViPhnHmNhiuBin2.HmnhiubinManyoftheideasdevelopedforfunctionsofonevariablealsoapplytofunctionsofmorethanonevariable.Inparticular,thefundamentalideaofderivativegeneralizesinaverynaturalwaytofunctionsofmorethanonevariables.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 7/791.Tphp RnRn= RR... R. .n= (x1, x2, ..., xn)[x1, x2, ..., xn RCho2imM(x1, x2, ..., xn)vM/(x/1, x/2, ..., x/n).Khongcchgia2imnycchobicngthc:d(M, M/) = _(x1x/1)2+ (x2x/2)2+ ... + (xnx/n)2n = 2 : R2= (x, y)[x, y R;d((x, y), (x0, y0)) = _(x x0)2+ (y y0)2n = 3 : R3= (x, y, z)[x, y, z R ;d((x, y, z), (x0, y0, z0)) = _(x x/0)2+ (y y0)2+ (z z0)2NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 8/792.HmnhiubinCho Rn.Gisvimix = (x1, x2, ..., xn)trong tanhnghacmtphntf (x)trongR,tanitaxcnhcmtnhxf : Rn Rvtanirnghmsf xcnhtrn vnhngitrtrongR.Khicgilminxcnhvf () = y = f (x)[x cgiltpnhcafTavitf : (x1, x2, ..., xn) z = f (x1, x2, ..., xn),haygnhnlz = f (x1, x2, ..., xn),trongx1, x2, ..., xncgilccbinclp,zcgilbinphthuc.Lu:nuhmf cchobicngthcnomkhngchrminxcnhthkhiminxcnhcahmf chiultpttcccgitrca (x1, x2, ..., xn)biuthcchocngha.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 9/79Biuthhmz = f (x, y)didngbiumitn(arrowdiagram)NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 10/793.th:Gf = _(x, f (x)) Rn+1[x _NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 11/79NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 12/79Biuthhmz = f (x, y)didngth(graph)Th d:f (x, y) = 6 3x 2yNguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 13/794.ngngtr-mtngtr:Chohmsf (x1, x2, ..., xn)viminxcnh .Tpttcnhngim(x1, x2, ..., xn)saochof (x1, x2, ..., xn) = c(c=constant) (+)cgiltpngtr(haycngilngngmc)Tphpccim (x, y)tha (+)gilngngtrcahmf (x, y)Tphpccim (x, y, z)tha (+)gilmtngtrcahmf (x, y, z)Lu:CcngngtrvmtngtrngviccgitrckhcnhauthkhnggiaonhauNguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 14/79Biuthhmz = f (x, y)didngngmc(contourcurve)NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 15/79NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 16/79Chng1.ViphnHmnhiubin3.CcmtbchaiCcmtbchaicphngtrnh:ax2+ by2+ cz2+ dxy + exz + f yz + gx + hy + i z + k = 0Mtsmtbchaichnhtcthnggp:MtEllipsoid:x2a2 +y2b2 +z2c2 = 1MtParaboloidElliptic:x2a2 +y2b2 = zMtParaboloidHyperbolic:x2a2 y2b2 = zNguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 17/79Chng1.Hmnhiubin4.Giihn,lintc1.ShitcadyimCho1dyimMk(xk1 , xk2 , ..., xkn ) RnvimM0(x01, x02, ..., x0n) Rn.Dy MkcgilhitnM0nud(Mk, M0) 0khik Nhvyrrngshittrong Rnlshittheota,nghalMk M0 = xk1 x01, xk2 x02, ..., xkn x0nkhik NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 18/792.Giihnhms:SthcLcgilgiihncahmf khiM M0nu\ > 0, > 0 : \M : 0 < d(M, M0) < = [f (M) L[ < Khitakhiu:L = limMM0f (M)Chthch:Cctnhchtvccphptnhshc(tng,hiu,tch,thng)cahmcgiihnutngtnhivihmmtbin.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 19/79Cthchotrnghphm2binnhsau:chohm2binf : Rv(x0, y0) R2.SthcLcgilgiihncahmsf (x, y)khi(x, y)tinv (x0, y0)nu \ > 0, > 0 :\(x, y) : 0 0saochoB

(M0) tacf (x0, y0) _ f (x, y) \(x, y) B

(M0)cci(aphng)tiM0nutnti> 0saochof (x0, y0) > f (x, y) \(x, y) B

(M0)NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 36/79iukincn(nhlFermat):Nuhmf tcctrtiM0(x0, y0)vnuf cccohmringtiM0thfx (x0, y0) =fy (x0, y0) = 0KhitaniM0limdng(criticalpoint).NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 37/79iukin:Chohm2binf : Rcccohmringncp2lintctiM0(x0, y0) vM0limdng.Tat:A =2fx2 (x0, y0); B =2fxy (x0, y0); C =2fy2 (x0, y0)= AC B2Khi,nu: < 0thf khngccctr(saddlepoint) > 0thM0limcctiunuA > 0hocC > 0;limccinuA < 0hocC < 0 = 0thtachacktlun,khitaphiquayvdngnhnghacctr(degeneratecase)NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 38/79Lu:ccihaycctiucamthmkhngnhtthitlimdng,dovycnkimtrangthitinhngimbinvtivccThd:f (x, y) = x + y +1xy ; x, y > 0NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 39/79Cctrciukin: tmcci/cctiucaf (x, y)tha(x, y) = 0Thd:tmimthucthuchyperbolxy = 3gnvigctanht?ThuttonnhntLagrange:XthmLagrangeL(x, y, ) = f (x, y) + (x, y)Bc1. Tmnghim (x0, y0, 0)cahsau:Lx(x, y, ) = Ly(x, y, ) = L(x, y, ) = 0Bc2.Xtviphncp2:d2L(x0, y0) = Lxxdx2+ 2Lxydxdy + Lyydy20 = /(x0, y0)dx + /(x0, y0)dyNu d2L(x0, y0) > 0 th f c cc tiu, nu d2L(x0, y0) < 0 th f c cc iNguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 40/79Chng1.ViPhnHmNhiuBin2.ohmtheohng-VectorGradientNguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 41/79Chng2.PhngTrnhViPhn1.MtvivdmuVd1:Mtvtckhilngmritdovilccncakhngkhtlvivntcri.Tnhvntcricavt?Giv(t)lvntcricavt.Khic2lctcnglnvt:trnglcF1 = mglccncakhngkh:F2 = v(t),trong > 0lhscnTheonhlut2Newtontac:ma = F = F1 + F2 = mg v(t)haytacthvitmdvdt = mg v(t):ylphngtrnhmngoihmcntmv(t),ncnchacohmv/(t).Phngtrnhnycgilphngtrnhviphncp1.Vytnhv,tacngiiphngtrnhviphncp1ny.Nutacois(t)lqungngiccavt,thtacphngtrnhviphncp2nhsau:md2sdt2 = mg dsdtNguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 42/79V d2:(Mhnhtngtrngdns)Chobittctngtrngdnstlvisdn.Hythitlpmhnhtngtrngdns?Gitlthigian(binclp)Plsdn(binphthuc)klhngstlTheogithittacphngtrnhsau:dPdt = kPhayP/(t) = kP(t) :ychnhlmhnhtngtrngdnscntm.Nghimcaphngtrnhnychnhl:P(t) = Cekt(C > 0)NucthmhnchrngsdnkhngvtquK,khitacphngtrnhsau:P/(t) = kP(t) (1 P(t)K)NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 43/79Chng2.PhngTrnhViPhn2.Phngtrnhviphncp1Phngtrnhviphnlmtphngtrnhchahmcntm,ohmcccpcanvbinclp.Cpcaphngtrnhviphnlcpcaonhtcaohmxuthintrongphngtrnh.TavitF(x, y(x), y/(x), ..., y(n)(x)) = 0ycgilphngtrnhviphnthngcpn.Tacthvitgnnhsau:F(x, y, y/, ..., y(n)) = 0NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 44/79Phngtrnhviphncp1lphngtrnhcdng(1) F(x, y, y/) = 0,Nutagiicphngtrnhiviy/thphngtrnhviphncp1cdng(2) y/= f (x, y) haydydx = f (x, y),trongf lmthmtheohaibinclp.Bi tonCauchy(haybi toniukinu)lbitontmnghimcaphngtrnhviphn(1)hoc(2)thaiukin(3) y(x0) = y0.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 45/79Nghimcaphngtrnhviphn(1)hoc(2)trnkhong lmthmsy =(x)xcnhtrn saochokhithayvo(1)hoc(2)tacngnhtthctrn :F(x, (x), /(x)) = 0, \x ,hoc/(x) = f (x, (x)), \x ,V d:XtbitonCauchyy/=yx, y(1) = 2.Ty/=yx, tacdydx =yxhaydyy=dxx.Lytchphnhaivtacln [y[ = ln [x[ + ln C1,(C1 > 0).Suyra y = Cx, x ,= 0, (C = C1).Vy(1) = 2 =C = 2.Vysuyranghimcabitonly = 2x,x ,= 0.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 46/79Nghimtngqut,nghimringvnghimkd:Nghimtngqut:Hmsy =(x, C)cgilnghimtngqutcaphngtrnhviphn(1)hoc(2)trongmin R2,nuvimiim (x0, y0) ,tntiduynhtmthngsC0saochoy =(x, C0)lnghimcaphngtrnhviphn(1)hoc(2)thaiukinuy(x0) = y0,nghal:tntiduynhtmthngsC0saochoi/y =(x, C0)lnghimcaphngtrnhviphn(1)hoc(2)trongkhongnochax0,ii/ y(x0, C0) = y0.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 47/79Nghimring:Taginghimringcaphngtrnhviphncpmtlnghimy =(x, C0)mtanhnctnghimtngquty =(x, C)bngcchchohngstyCmtgitrcthC0.Nghimkd:NghimkhngthnhnctnghimtngqutchodClybtkgitrnoscgilnghimkd.Khigii(1)hoc(2)ckhitacngkhngnhncnghimtngqutdidngtngminhy =(x, C),mnhncmththccdng(4) (x, y, C) = 0.Khinghimtngqutcxcnhdidnghmnv(4)cgil tch phn tng qutca (1) hoc (2). Cho C = C0ta c phng trnh(5) (x, y, C0) = 0.mtagiltchphnringcaphngtrnhviphnnitrn.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 48/79Chng2.PhngTrnhViPhnCcdngcaphngtrnhviphncp1:dngtchbinPhngtrnhviphncp1tchbinlphngtrnhviphncdng(6) f (x)dx + g(y)dy = 0.Cchgii:Lytchphnbtnhhaivtactchphntngqut_f (x)dx + _g(y)dy = C.V d:Giiphngtrnhviphnx1+x2dx +y1+y2dy = 0.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 49/79Chthch:Phngtrnhdngsauycthavdng(6)nhsau:(7) f1(x)g1(y)dx + f2(x)g2(y)dy = 0Nug1(y) = 0tiy = b(tclg1(b) = 0)thy = blnghimca(7).Nuf2(x) = 0tix = a(tcl f2(a) = 0)thx = alnghimca(7).Nug1(y)f2(x) ,= 0thchiahaivca(7)chog1(y)f2(x),tacf1(x)f2(x)dx +g2(y)g1(y)dy = 0.Trongtrnghpny,lytchphnhaivtactchphntngqut_f1(x)f2(x)dx + _g2(y)g1(y)dy = C.V d:Giiphngtrnhviphny/= xy(y + 2).NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 50/79Chthch:Phngtrnhdngsauycngcthavphngtrnhviphntchbinbngcchiquanhmmiz = ax + by + c.(8) y/= f (ax + by + c)Thtvydzdx = a + bdydx = a + bf (z)Nua + bf (z) = 0khiz = z0,thax + by + c = z0lnghimca(8).Ccnghimkhctmcbngcchchiahaivcaphngtrnhchoa + bf (z)rilytchphn,tac_dza+bf (z) = x + C.V d:Giiphngtrnhviphn y/= (x y + 1)2.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 51/79Chng2.PhngTrnhViPhnCcdngcaphngtrnhviphncp1:dngngcpcp1Phngtrnhviphnngcpcp1lphngtrnhcdng(9) y/= f (yx ),Cchgii:Taavphngtrnhviphntchbinbngcchiquanhmmiu =yx.Khiy = ux, y/= xu/+ u.Thayvo(9)tacxu/+ u = f (u).hayx dudx = f (u) u.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 52/79Xt2trnghp:f (u) u ,= 0,tacdxx=duf (u)u.Tchphnhaivcaphngtrnhny,tacln [x[ = _duf (u)u + ln [C[ = (u) + ln [C[ ,trong (u) = _duf (u)u.Dox = Ce(u).Vytchphntngqutca(9)lx = Ce(y/x)NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 53/79f (u) u = 0,thf (y/x) = y/xvphngtrnh(9)cdngdydx =yxhaydyy=dxxTchphnhaivca,tacln [y[ = ln [x[ + ln [C1[ ,hayy = Cx, x ,= 0(C = C1)lnghimtngqutca(9)trongtrnghpf (y/x) = y/x.Hocnuf (u) u = 0,tiu = u0thbngcchthtrctiptathyrnghmy = u0x, (x ,= 0)cnglnghimcaphngtrnhcho.V d:Giiphngtrnhviphnngcp y/=x+yxy =1+yx1yx.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 54/79Chng2.PhngTrnhViPhnCcdngcaphngtrnhviphncp1:dngtuyntnhcp1Phngtrnhviphntuyntnhcp1lphngtrnhcdng(10) y/+ p(x)y = q(x),trongp(x),q(x)lcchmslintcchotrc.Nuq(x) = 0,th(10)cgilphngtrnhviphntuyntnhcp1thunnht.Nuq(x) ,= 0,th(10)cgilphngtrnhviphntuyntnhcp1khngthunnht.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 55/79Phngphpbinthinhngs(Lagrange)Trchttaxtphngtrnhviphntuyntnhcp1thunnhttngngvi(10)(11) y/+ p(x)y = 0, haydydx = p(x)y.Viy ,= 0,tacdyy= p(x)dx.Lytchphnhaiv,tacln [y[ = _p(x)dx + ln C1.Do(12) y = Ce_p(x)dx(C = C1)lnghimtngqutcaphngtrnh(11).Viy = 0cnglnghimcaphngtrnh(11)vcnglmtnghimringcaphngtrnh(11)ngviC = 0.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 56/79BygitacoiCkhngphilhngsmlmthmkhvitheobinx.TastmhmsC = C(x)biuthcy = C(x)e_p(x)dxlnghimcaphngtrnhkhngthunnht(10).Lyohm(12),tacy/= C/(x)e_p(x)dxp(x)C(x)e_p(x)dx.Thayvophngtrnh(10),tacC/(x)e_p(x)dx= q(x), hay dC = q(x)e_p(x)dxdxLytchphnhaiv,tacC = C(x) = _q(x)e_p(x)dxdx + C1.Donghimtngqutcaphngtrnhtuyntnhkhngthunnht(10)l(13) y = e _p(x)dx__q(x)e_p(x)dxdx + C1_.V d:Giiphngtrnhviphn y/+ y cos x = esin x.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 57/79PhngphpBernoulliTastmnghimcaphngtrnhkhngthunnht(10)didng(14) y = u(x)v(x).Thvo(10),tacu/v + uv/+ p(x)uv = q(x),hay(15) [u/+ p(x)u]v + uv/= q(x).Chnu(x)lmtnghimcaphngtrnh(16) u/+ p(x)u = 0,tac(17) u = e_p(x)dxT(15)(17),tac v/e_p(x)dx= q(x).Vyv(x) = _q(x)e_p(x)dxdx + C1.Cuicngtanhncnghimtngqutchobi(13).V d:Giiphngtrnhviphny/sin x y cos x = sin2xx2.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 58/79PhngphpthastchphnNhnhaivca(10)vithase_p(x)dx,tacy/e_p(x)dx+ p(x)e_p(x)dxy = q(x)e_p(x)dx,mvtricangthcnychnhlohmcatchsye_p(x)dx.Vytavitlingthcnynhsauddx_ye_p(x)dx_ = q(x)e_p(x)dx.Lytchphnhaiv,tacye_p(x)dx= _q(x)e_p(x)dxdx + C.Vynghimtngqutca(10)ly = e _p(x)dx__q(x)e_p(x)dxdx + C_.V d:Giiphngtrnhviphn y/+ 2xy = 4x.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 59/791.PhngTrnhViPhnCcdngcaphngtrnhviphncp1:phngtrnhBernoulliPhngtrnhviphnBernoulli lphngtrnhcdng:(18) y/+ p(x)y = q(x)y,trongp(x),q(x)lcchmslintccaxchotrcvlmthngsthcchotrc.Vi = 0,th(18)lphngtrnhviphntuyntnhcp1khngthunnht.y/+ p(x)y = q(x).Vi = 1,th(18)lphngtrnhviphntuyntnhcp1thunnht.y/+ [p(x) q(x)]y = 0.Dovyytachcnxt ,= 0v ,= 1.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 60/79Cchgii:Gis ,= 0v ,= 1.Nu > 0,thy = 0lmtnghimringca(18).Ngclinu _ 0,thy = 0khnglnghimca(18).Gisy ,= 0,chiahaivca(18)choy,tacyy/+ p(x)y1= q(x).t z = y1, ta c z/= (1 )yy/v phng trnh trn c vit li(19) z/+ (1 )p(x)z = (1 )q(x).ylphngtrnhviphntuyntnhcp1ivinhmz.Saukhigiitmcnghimtngqutcaphngtrnh(19),tatrvnybicngthcz = y1,tacnghimtngqutcaphngtrnh(18).V d:Giiphngtrnhviphny/2xy = 4x3y2.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 61/79Chng2.PhngTrnhViPhn3.Phngtrnhviphncp2Phngtrnhviphncp2lphngtrnhcdng(20) F(x, y, y/, y//) = 0,Nugiicphngtrnh (20)iviy//thphngtrnhviphncp2cdng(21) y//= f (x, y, y/),trongf lmthmchotrctheobabinclp.Nghimcaphngtrnhviphn(20)trnkhong lmthmsy =(x)xcnhtrn saochokhithayvo(20)tacngnhtthctrn :F(x, (x), /(x), //(x)) = 0, \x .NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 62/79Phngtrnhviphncp2cnghimphthucvohaihngs,nnxcnhmtnghimcthcnchaiiukinno.NgitathngxtbitonCauchy(bitoniukinu).Bi tonCauchylbitontmnghimcaphngtrnhviphncp2thaiukinu(22) y(x0) = y0, y/(x0) = y/0 .vix0,y0,y/0lnhngschotrc.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 63/79nhl(vstntivduynhtnghim) Nuhmsf (x, y, y/)lintctrongminmnocha (x0, y0, y/0 ),tntinghimcabiton(21),(22).Hnna,nufy,fy/lintcvbchntrongminmnocha(x0, y0, y/0 ),thnghimylduynht.NghimcaphngtrnhviphncphaithngphthucvohaihngsthcC1, C2,vcdngy =(x, C1, C2)NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 64/79Nghimtngqut,nghimringHms y =(x, C1, C2)cgilnghimtngqutcaphngtrnhviphncphai (21)trongmin R3nu (x0, y0, y/0 ) ,tntiduynhtmtcphngs (C01 , C02 )saochoy =(x, C01 , C02 )lnghimcaphngtrnhviphn (21)thacciukinuy(x0) = y0, y/(x0) = y/0 .Nghimnhnctnghimtngquty =(x, C1, C2)bngcchcho cc hng s C1, C2nhng gi tr c th c gi l nghim ring.Phngtrnh(x, y, C1, C2) = 0chotamiquanhgiabinclpvnghimtngqutcaphngtrnhviphncphaicgiltchphntngqutcantrn.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 65/79NuchoC1 = C01,C2 = C02lnhnggitrcthtacphngtrnh(x, y, C01 , C02 ) = 0mtaginltchphnnghimringcaphngtrnhviphnnitrn.Vphngdinhnhhc,tchphntngqutcaphngtrnhviphncphaixcnhmthngcongtrongmtphngtaphthucvohaithamsty.Ccngcongycgilngcongtchphncaphngtrnhviphn.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 66/79Chng2.PhngTrnhViPhnCcdngcaphngtrnhviphncp2:dnggimcpcXtphngtrnhviphncphaicdngy//= f (x, y, y/)mtacthachngvcpmt.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 67/79Phngtrnhvi phndngy//= f (x)(23) y//= f (x)Cchgii.V y//= (y/)/nnt (23)tac y/= _f (x)dx + C1.Lytchphnmtlnna,tacy/= _ __f (x)dx_dx + C1x + C2.trongC1, C2lcchngsty.V d.Tmnghimtngqutvnghimringcaphngtrnhviphny//= sin xthacciukinuy(0) = 0, y/(0) = 1.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 68/79Phngtrnhvi phndngy//= f (x, y/)(24) y//= f (x, y/)Cchgii.t y/= p,khiy//= p/v (24)cdngp/= f (x, p).ylphngtrnhviphncp1.Nugiic,tacnghimtngqutlp =(x, C1).Vy/= p,nntacy/=(x, C1).Suyranghimtngqutcaphngtrnh (24)ly = _(x, C1)dx + C2.V d.Giiphngtrnhviphny//= x y/x .NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 69/79Phngtrnhvi phndngy//= f (y, y/)Cchgii.t y/= p = p(y)vxemnhlhmcay.Lyohmhaivcangthcnytheox,tacy//=dpdx =dpdydydx = pdpdy .Khi,phngtrnhchocdngp/ dydx = f (y, p).lphngtrnhviphncp1vinhmlp = p(y).Nuphngtrnhnygiic,tacp =(y, C1),haydydx =(y, C1),dy(y,C1) = dx.Donghimtngqutcaphngtrnhchol y/=(x, C1).Suyratchphntngqutcaphngtrnhchol_dy(y,C1) = x + C2.V d.Giiphngtrnhviphnyy//y/2= 0.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 70/79Chng2.PhngTrnhViPhnCcdngcaphngtrnhviphncp2:phngtrnhtuyntnhcp2hshngPhngtrnhviphntuyntnhcp2chshnglphngtrnhcdng(25) y//+ py/+ qy = f (x), a < x < b,trongp,qlcchngs.Talungithitf (x)lhmlintctrongkhong (a, b).Nuf (x) = 0,phngtrnh(26) y//+ py/+ qy = 0,cgilphngtrnhthunnhttngngviphngtrnh(25).NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 71/79Bi tonCauchy:Xtbitontmnghimcaphngtrnh(25)thaiukinu(27) y(x0) = y0, y/(x0) = y/0 ,vi x0 (a, b),y0,y/0chotrc.nhl(stnti vduynhtnghim):Nuhmsf (x)lintctrongkhong (a, b),thvimi x0 (a, b),vvimiy0,y/0chotrc,bitonCauchy (25), (26)cduynhtmtnghim.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 72/79Phngtrnhvi phnthunnhtXtphngtrnhviphnthunnht(28) y//+ py/+ qy = 0.nhl:Choy1(x)v y2(x)lhainghimclptuyntnhtrong(a, b)caphngtrnhthunnht (28).Khinghimtngqutcaphngtrnh (28)cdng(29) y = C1y1(x) + C2y2(x),viC1, C2lhaihngs.Nhvymuntmnghimtngqutcaphngtrnhthunnht (28),tachcntmhainghimringclptuyntnhcan,rilythptuyntnhcachng.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 73/79Tatmnghimringca (28)didng(30) y = ekx,trongklmthngsno.Tacy/= kekx,y//= k2ekx.Thayccbiuthcy,y/,y//vo (28)tacekx(k2+ pk + q) = 0.Vekx,= 0nntac(31) k2+ pk + q = 0.Vynukthamnphngtrnh (31)thhmy = ekxlmtnghimringcaphngtrnh (28).Phngtrnh (31)cgilphngtrnhctrngcaphngtrnhviphn (28).NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 74/79CbatrnghpsauyPhngtrnh (31)chainghimthcphnbitk1,k2.Khitachainghimringcaphngtrnh (28)ly1 = ek1x, y2 = ek2x.Hainghimyclptuyntnh,vy1y2= e(k1k2)x,=hngs.Donghimtngqutcaphngtrnh(28)ly = C1ek1x+ C2ek2xtrongC1, C2lcchngsty.V d.Giiphngtrnhviphny//6y/+ 8y = 0.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 75/79Phngtrnh (28)cnghimkpk1 =k2 = p/2.Khitacy1 = ek1xvy2 = xek1xl2nghimca (28).Vynghimtngqutcaphngtrnh (28)ly = C1ek1x+ C2xek1x= (C1 + C2x)ek1x,trongC1, C2lcchngsty.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 76/79Phngtrnh (31)chainghimphclinhpk1 = + i ,k2 = i .Tachainghimringcaphngtrnh(28)ly1 =y1+y22= excos x,y2 =y1y22i= exsin x,Nghimtngqutcaphngtrnh(31)ly = C1excos x + C2exsin x= ex(C1 cos x + C2 sin x),trongC1, C2lcchngsty.V d.Giiphngtrnhviphny//+ 4y/+ 4y = 0.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 77/79Phngtrnhvi phnkhngthunnhtBygitaxtphngtrnhviphntuyntnhkhngthunnhtsau(32) y//+ py/+ qy = f (x), a < x < b,trongp,qlcchngsvf (x)lhmlintctrongkhong (a, b).Xtphngtrnhviphnthunnhttngngvi (32)(33) y//+ py/+ qy = f (x).nhl. Nghim tng qut ca phng trnh khng thun nht (32)bngtngcanghimtngqutcaphngtrnhthunnhttngng (33)vimtnghimringnocaphngtrnhkhngthunnht (32).NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 78/79Tachxttrnghpf (x) = exPn(x)tronglsthc,Pn(x)lathcbcn.Nu khng l nghim ca (31), th ta tm nghim ring yrtheo dngyr = exQn(x),viQn(x)lathcbcnvin + 1hschabit.tmcchschabit,tathayyrvophngtrnh (32)ringnhtcchscacclythacngbccaxhaivtascmth (n + 1)phngtrnhbcnhtvi (n + 1)nlcchscaathcQn(x).Nulnghimncaphngtrnhctrng (31),thtatmnghimringyrtheodngyr = xexQn(x),viQn(x)lathcbcn.Nulnghimkpcaphngtrnhctrng (31),thtatmnghimringyrtheodngyr = x2exQn(x),viQn(x)lathcbcn.V d.Tmnghimtngqutcaphngtrnhy//y/2y = 4x2.NguynTh ThuVn-TrnVKhanh(HCMUNS) Gii TchC2 3/2010 79/79