Сборник задач по высшей математике 2 том
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2.1.33. PewHTb ,n;H<p<pepeHII.Ha.JIbHoe yprumeHHe:
a) dy = 2cosx; |
6) siny' = 1. |
dx |
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2.1.34. IIpH KaKOM 3Ha'IeHHHC 3a,n;aHHruI <PYHKII.HH HBJIHeTCH peweHHeM
,n;aHHoro ypaBHeHHH: |
6) y = x 3 , y' = Cx2. |
a) s = Ct+4, s' = -1; |
2.1.35. HanHcaTb ypaBHeHHe reOMeTpH'IeCKOrOMeCTa TO'IeK(x, y), HBJIHIOIII.HXCH TO'IKaMHMaKCHMYMa HJIH MHHHMyMa peweHHit ypaBHeHHH
y'=f(x,y).
2.1.36. KaK ,n;OKa3aTb, 'ITOxy+ln ~ = C eCTb 06III.Hit HHTerpa.JI ypaBHeHHH
x(1 + xy)y' = y(1 - xy)?
2.1.37. 3HruI, 'ITOY = Clnx HBJIHeTCH 06III.HM peweHHeM ypaBHeHHH
xy'lnx = y,
HaitTH HHTerpa.JIbHYIO KPHBYIO, npOXOAHIIJ.Y1O 'Iepe3TO'IKYM(e, 1). 2.1.38. KaKruI H3 <PYHKII.Hit:
1 |
y = |
Jln(x + 1) |
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y=x+l' |
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HBJIHeTCH peweHHeM ,n;H<p<pepeHII.Ha.JIbHOrO ypaBHeHHH
d - |
dx ? |
YY - 2(x + 1)'
2.1.39.PewHTb ypaBHeHHH:
a) 2y' = 0; |
6) y' = x; |
B)y' = y.
2.1.40.KaKHe H3 npHBe,n;eHHblx ypaBHeHHit HBJIHIOTCH ypaBHeHHHMH C pa3- ,n;eJIHIOIII.HMHCH nepeMeHHbIMH?
a) y' = |
3y -1; |
6) xdy + ydx = y2 dx; |
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B) (1 - |
x 2)y' + xy =,1; |
r) xy' + y = cosy; |
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.n.) y' = (x + y)2; |
e) y' + x 2y = eX; |
= l' |
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lK) y' - |
xy2 = 2xy; |
3) e-Y (1 + dY ) |
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u) x 2y' -1 = cos2y; |
dx |
' |
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K) y = xeY'. |
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Pew,umb iJutPtPepe'H:qua.llbH'bte ypa6HeHUSI:
2.1.41. (..jXY + y'x)y' - Y = O.
, y+l
2.1.43. y = x+ l'
,
2.1.45.y% + eY = O.
2.1.47.y' + y = 5.
2.1.49.dy - Y cos2 X dx = O.
2.1.51.(eX + l)eY y' + eX(1 + eY )
,xsinx 0
2.1.52.y + ycosy = .
2.1.42.
2.1.44.
2.1.46.
2.1.48.
2.1.50. = O.
y' = 3x - y •
ds + stgtdt = O.
x + xy + y' (y + xy) = O. v' - 4tv = O.
, |
. x-y . x+y |
y |
= sm - 2 - - sm -2-' |
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2.1.53. y' = cos(y - x). (TIOJIO)KHTb y - X = t.)
2.1.54. (xy + x)dx = 1. dy
2.1.55.6xdx - 6ydy - 2x2ydy + 3Xy2 dx = o.
2.1.56. |
X2dy + (y - a) dx = O. |
2.1.57. |
y' tg x - |
Y = a. |
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2.1.58. |
y' cos x - (y + 1) sin x = O. |
2.1.59. |
y' - 2yctgx = ctgx. |
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2.1.60. |
y - |
xy' = 1 + x 2y' . |
2.1.61. |
dx |
dy |
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( |
)= ( |
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xy-1 |
yx+2 |
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y |
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- |
y In3 y |
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2x + 2xy2 + ../2 - |
x2y' = O. |
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2.1.62. |
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v'x+I. |
2.1.63. |
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Ha11,mu "tacm'H.'bte pewe'H.'I.J.SI autjjtjjepe'H.v,ua.ll,'b'H.'btX ypa6'H.e'H.u11,:
2.1.64.x 2dy - y2 dx = 0, y G) = ~.
2.1.65.1 + y2 = xyy', y(2) = 1.
2.1.66.(x + xy2) dx + (x 2y - y) dy = 0, y(O) = 1.
2.1.67.Y'(x 2 - 2) = 2xy, y(2) == 2.
2.1.68.cosxsinydy = cosysinxdx, Y(1I") = 11".
2.1.69.y' = 1,5 \!y, y(-2) = 1.
2.1.70.y' = 2x +y + 2x - y , y(O) = o.
2.1.71. |
xy' - |
Y |
0, y(e) = |
1. |
- I = |
nx
2.1.72.y' sin x - (2y + 1) cos x = 0, y (i) = 1.
2.1.73.(eX + 8) dy - yeXdx = 0, y(O) = 1.
2.1.74.RaitTH KPHBYIO, npOXO.IVlIIJ.YIO qepe3 TOqKY A(2, 16), 3HaJI, qTO yr-
JIOBOit K09ci>ci>HD:HeHT KaCaTeJIbHoit B JII060it TOqKe KPHBOit:
a) B TpH pa3a 60JIblne yrJIOBOrO K09ci>ci>HD:HeHTa np1lMoit, coe.n;HH1IIOmeit 9Ty )Ke TOqKY c HaqanOM Koop.n;HHaT,
6) paBeH KBa.n;paTY op.n;HHaTbI 9TOit TOqKH.
2.1.75. RaitTH ypaBHeHHe KPHBOit, npOXo.IVlmeit qepe3 TOqKY A(4, 1), ,n;JI1I KOTOPOit:
a) OTpe30K JII060it KacaTeJIbHoit K KPHBOit, 3aKJIIOqeHHblit Me)K.n:y OC1lMH Koop.n;HHaT, .n;eJIHTC1l TOqKOit KacaHH1I nOnOJIaM;
6) OTpe30K KaCaTeJIbHoit Me)K.n:y TOqKOit KacaHH1I 1'1 OCbIO a6CD:HCC
.n;eJIHTC1I nOnOJIaM B TOqKe nepeCeqeHH1I C OCbIO op.n;HHaT.
2.1.76.IIolmacame.ll,'b'H.o11, KPHBOit y = f(x) B TOqKe M Ha3bIBaeTC1I npo-
eKn:H1I AP Ha OCb Ox OTpe3Ka AM KacaTeJIbHoit K 9TOit KPHBOit, r.n;e A TOqKa nepeCeqeHH1I KaCaTeJIbHoit C OCbIO Ox (pHC. 5) RaitTH ceMeitcTBo KPHBbIX, y KOTOPbIX nO,n;KaCaTeJIbHaJI HMeeT .n;JIHHY, paBHYIO 2.
2.1.77. RaitTH KPHBYIO, npOXO.IVlIIJ.YIO qepe3 TOqKY A(l, 1), .n;JI1I KOTOPOit nJIOma.n;b TpeyrOJIbHHKa, 06pa30BaHHOro KaCaTeJIbHoit, op.n;HHaToit TOqKH KacaHH1I 1'1 OCbIO a6CD:HCC, paBHa 1.
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Puc. 5 |
2.1.18. |
HaiiTH KPHBYIO, Y KOTOpoii cYMMa ,1I,JUIH KacaTeJIbHOii (TOqHee, |
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)I,JIHHbI ee OTpe3Ka OT TOqKH KacaHH.H ,n:o TOqKH nepeCeqeHH.H C |
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OCbIO a6c[(Hcc) H nO,n:KacaTeJIbHOii B JlIo60ii ee TOqKe palma npOH3- |
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Be,n:eHHIO Koop,n:HHaT TOqKH KacaHH.H. |
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2.1.19. |
CKOPOCTb pacna,u.a pa,u.H.H npOnOp[(HOHaJIbHa HaJIHqHOii ero Macce. |
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Onpe,n:eJIHTb, qepe3 CKOJIbKO JIeT OT 1 Kr pa,II.H.H OCTaHeTC.H 0,7 Kr, |
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ecJIH H3BeCTHO, qTO nepHO,n: nOJIypacna,u.a pa,u.H.H (BpeM.H, 3a KOTo- |
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poe Macca pa,u.H.H YMeHblIIaeTC.H B,n:Boe) paBeH 1590 JIeT. |
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2.1.80. |
CKOPOCTb pa3MHO)KeHH.H HeKoTopbIX 6aKTepHii npOnOp[(HOHaJIbHa |
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KOJIHqecTBY 6aKTepHii, HMeIOIIIHXC.H B HaJIHqHH B paccMaTpHBae- |
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MbIii MOMeHT BpeMeHH t. KOJIHqeCTBO 6aKTepHii 3a 4 qaca YTPOH- |
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JIOCb. HaiiTH 3aBHCHMOCTb KOJIHqeCTBa 6aKTepHii OT BpeMeHH, eCJIH |
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npH t = 0 HX 6bIJIO a. |
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2.1.81. |
MOTOPHa.H |
JIO,n:Ka ,n:BH)KeTC.H B cnoKoiiHoii Bo,n:e co CKOPOCTbIO |
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20 KM/qac. qepe3 O,n:HY MHHYTY nOCJIe BbIKJIIOqeHH.H ,n:BHraTeJI.H |
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ee CKOPOCTb YMeHblIIHJlacb ,n:o 2 KM/qac. Onpe,n:eJIHTb CKOPOCTb |
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JIO,n:KH qepe3 ,n:Be MHHYTbI nOCJle OCTaHOBKH ,n:BHraTeJI.H, CqHTa.H, |
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qTO conpOTHBJIeHHe BO,n:bI npOnOp[(HOHaJIbHO CKOPOCTH ,n:BH)KeHH.H |
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JIO,n:KH. |
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2.1.82. |
MeTaJIJIHqecKa.H 6oJIBaHKa, HarpeTa.H ,n:o 420°C, OXJla)K,n:aeTC.H B |
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B03.n:yxe, TeMnepaTypa KOTOPOro 20°C. Qepe3 15 MHHYT nOCJle Ha- |
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qaJIa OXJla)K,n:eHH.H TeMnepaTypa ,n:eTaJIH nOHH3HJIacb ,n:o 120°C. |
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Onpe,n:eJIHTb TeMnepaTYPY 60JIBaHKH qepe3 30 MHHYT OXJla)K,n:e- |
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HH.H, CqHTa.H, qTO CKOPOCTb OXJla)K,n:eHH.H npOnOp[(HOHaJIbHa Pa3- |
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HOCTH Me)K.n:y TeMnepaTypoii TeJIa H TeMnepaTypoii B03.n:yxa. |
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2.1.83. |
IIPH 6pO)KeHHH CKOPOCTb npHpocTa ,n:eiicTBYIOIIIero <ilepMeHTa npo- |
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nOp[(HOHaJIbHa ero KOJIHqeCTBY. Qepe3 tl qacOB nOCJIe HaqaJIa |
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6pO)KeHH.H |
Macca <ilepMeHTa COCTaBHJIa ml r, a qepe3 t2. qacOB |
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(t2 > tt) - |
m2 r (m2 > mt). KaKoBa 6bIJIa nepBOHaqaJIbHa.H Macca |
<ilepMeHTa?
2.1.84. BpaIIIaIOI[(Hiic.H B )KH,n:KOCTH ,n:HCK 3aMe)I,JI.HeT CBoe ,n:BH)KeHHe no,n: ,n:eiicTBHeM CHJIbI TpeHH.H, npOnOp[(HOHaJIbHOii yrJIoBoii CKOPOCTH
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BpameHH.H w. IbBeCTHO, qTO ,n:HCK, HaqaBIIIHiI: BpamaTbC.H co CKOPoCTbIO 18 06/c, IIO HCTeqeHHH 45 C BpamaeTC.H co CKOPOCTbIO 6 06/c. C KaKoil: yrJIOBoil: CKOPOCTbIO 6y,n:eT BpamaTbC.H ,n:HCK IIO HCTeqeHHH 90 C IIOCJIe HaqaJIa 3aMe)I.JIeHH.H? B KaKOil: MOMeHT BpeMeHH W
6y,n:eT paBH.HTbC.H 1 06/c?
KOHTponbHble BonpocbI III 60nee CnO)l(Hbie 3aAa-'1II
2.1.85. MorYT JIH HHTerpaJIbHble KpHBble ,n:H<p<pepeHl:.J;HaJIbHOrO ypaBHe-
HH.H yl = f(x) IIepeceKaTbC.H?
2.1.86.MO:lKHO JIH MHO:lKeCTBO Bcex peIIIeHHiI: ypaBHeHH.H yl = Y IIpe,n:CTaBHTb B BH,n:e:
a) y = Cex ;
0) y = VCex ;
6)y=C1 ex +C2 ; r) y = sin C . eX;
e) y = .1 eX?
C
2.1.87.B pe3epByape HaxO,n:HTC.H 80 JI pacTBopa, co,n:ep:lKamero 8 Kr co-
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JIH. Ka:lK.n:yIO MHHyTy B Hero BJIHBaeTC.H 4 JI BO,n:hI H BbITeKaeT 4 JI |
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pacTBopa, IIpH 9TOM KOH~eHTpa~H.H COJIH IIo.n:.n:ep:lKHBaeTC.H paB- |
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HOMepHoil: (IIYTeM IIepeMeIIIHBaHH.H). CKOJIbKO COJIH OCTaHeTC.H B |
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pe3epByape qepe3 40 MHHYT? |
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2.1.88. |
CKOPOCTb HCTeqeHH.H BO,n:hI H3 cocy,n:a qepe3 MaJIOe OTBepCTHe OIIpe- |
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,n:eJI.HeTC.H <P0PMYJIOil: v = 0,6y'2gh, r,n:e h - |
BblCOTa cToJI6a :lKH,n:- |
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KOCTH Ha,n: OTBepCTHeM, 9 - YCKopeHHe cB060,n:Horo IIa,n:eHH.H (g ~ |
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~ 10 M/C2 ). 3a KaKoe BpeM.H BbITeqeT BC.H Bo,n:a H3 |
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a) 3aIIOJIHeHHOro IIOJIyc<pepHqeCKOrO KOTJIa ,n:HaMeTpa 2 M qepe3 |
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KpyrJIoe oTBepcTHe Ha ,n:He 0,1 M; |
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6) ~HJIHH,n:pHqeCKoro 6aKa pa,n:uyca R = 0,5 M H BbICOTOil: H = 2 M |
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qepe3 KpyrJIOe OTBepCTHe B ,n:He pa,n:Hyca r = 0,02 M. |
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2.1.89. TeJIO ,n:BIDKeTC.H IIO IIP.HMOil: co CKOPOCTbIO, |
06paTHo IIpOIIOp~o |
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HaJIbHOil: IIpoil:,n:eHHoMY "YTH. B HaqaJIbHbliI: MOMeHT TeJIO HMeJIO |
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CKOPOCTb Vo = 15 M/C H HaxO,n:HJIOCb Ha pacCTO.HHHH 4 M OT HaqaJIa |
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OTCqeTa IIyTH. OIIpe,n:eJIHTb CKOPOCTb TeJIa qepe3 8 C IIOCJIe HaqaJIa |
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,n:BIDKeHH.H. |
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2.1.90. |
Cy,n:HO Bo,n:OH3MemeHHeM 10000 TOHH ,n:BIDKeTC.H IIP.HMOJIHHeil:Ho co |
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CKOPOCTbIO 10 M/C. COIIpOTHBJIeHHe BO,n:hI IIpOIIOp~HOHaJIbHO KBa- |
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,n:paTy CKOPOCTH cy,n:Ha H paBHo 20000 H IIpH CKOPOCTH 1 M/C. Ka- |
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Koe paccTO.HHHe IIpoil:,n:eT Cy,n:HO IIOCJIe BbIKJIIOqeHH.H ,n:BHraTeJI.H, |
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IIpe:lK,n:e qeM ero CKOPOCTb YMeHbIIIHTC.H ,n:o 2 M/C? |
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2.1.91. |
PeIIIHTb ypaBHeHHe 2 ch y dx = (v'x+1 v'x=l)dy. |
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2.1.92. |
PeIIIHTb ypaBHeHH.H: |
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a) yl |
= ysinx2 ; |
2y + 3) dy = 0 (IIOJIO:lKHTb 2x - Y = t),; |
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6) (2x - y) dx + (4x - |
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cos Y - sin y - |
1 |
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0) y |
= cosx-sinx+l; |
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r) ~dx + Vl- x2dy = 0, y(O) = 1;
.n.) y' = 3x - 2y + 1 (nOJIO)l{HTb 3x - y + 1 = t);
e)y' = cos(y - x).
§2. OAHOPOAHblE AVI«lJ«lJEPEHU.VlAllbHbIE YPABHEHVlSI
~CPYHKIJ;HH !(x, y) Ha3hIBaeTCH OOHopoihwiJ, ,pywlCv,ueiJ, cmeneHU n, r.n:e n-IJ;eJIoe,
eCJIH rrpH JIIo60M a HMeeT MeCTO TOlK.n:eCTBO !(ax, ay) = an !(x, y). |
~ |
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B '1acTHOCTH,<PYHKIJ;HH !(x, y) - O.n:Hopo.n:HaH HYJIeBoil: CTerreHH, eCJIH |
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!(ax, ay) = !(x, y). |
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P(x, y) dx + Q(x, y) dy = 0 |
(2.1) |
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Ha3hIBaeTCH OOHOPOOH'bI..M, eCJIH P(x, y) H Q(x, y) - O.n:Hopo.n:HhIe <PYHKIJ;HH o.n:HHa- |
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KOBOil: CTerreHH. |
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YpaBHeHHe (2.1) MOlKeT 6hITb rrpHBe.n:eHO K BH)l;y |
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y' |
=! (~). |
(2.2) |
O.n:Hopo.n:Hoe ypaBHeHHe rrpeo6pa3yeTcH B ypaBHeHHe C pa3.n:eJIHIOIn;HMHCH rre- |
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peMeHHhIMH rrpH rrOMOIn;H 3aMeHhI rrepeMeHHoil: |
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!!.=u |
T.e. y =ux, |
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x |
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r.n:e u = u(x) - |
HOBaH HeH3BeCTHaH <PYHKIJ;HH (MOlKHO TaKlKe rrpHMeHHTb rro.n:CTa- |
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HOBKY ~ = u). |
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3aMe'iaHue. |
YpaBHeHHe BH.n:a y' = |
ax+by+c |
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b |
rrpHBo.n:HTCH K O.n:HOPO.n:HOMY |
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alx+ |
lY+Cl |
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C rrOMOIn;bIO 3aMeH x = u + a, y = v + /3, |
r.n:e a |
H /3 - '1HCJIa,KOTophIe rro.n:6HpaIOT |
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COOTBeTCTBYIOIn;HM o6pa30M (CM. 3a.n:a'lY2.2.5). STOT lKe rrpHeM HCrrOJIb3yeTcH rrpH
pemeHHH ypaBHeHHiI: BH.n:a y |
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=! |
(ax+bY+c) |
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b |
lY + Cl |
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alX + |
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2.2.1.IIpOHHTerpHpOBaTb CJIe.n.ylOmHe .n:H<p<pepeHU;HaJIbHble ypaBHeHHlI:
a) (y2 + xy) dx - x2 dy = 0; |
xy2 |
yx2 |
6)y'= |
3 ,y(-I)=I; |
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y |
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x |
B) xy' - Y + xe x = O. |
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Q a) 3a.n;aHHoe ypaBHeHHe HMeeT BH.n: (2.1). K03<P<PHu;neHTbI npH dx 1'1 dy, T. e. P(x, y) = y2 +xy 1'1 Q(x, y) = -x2, lIBJIlIlOTClI O.n:HOPO.n:HblMH <PYHKU;HlIMH o.n:Hofi 1'1 Tofi )l{e CTeneHH (BTOpofi). ,ll;eficTBHTeJIbHO,
P(o.x,o.y) = (o.y)2 + (o.x .o.y) = 0.2(y2 + xy) = 0.2 P(x, y),
64
Q(ax, ay) = -(ax)2 = a 2(_X2) = a 2Q(x, y), n = 2. CJIe,Il;OBaTeJIbHO, ,Il;aJlHoe
ypaBHeHHe O,Il;HOPO,Il;Hoe. TIOJIOlKHM y = ux. Tor,Il;a dy = xdu+udx, H ,Il;aHHoe ypaBHeHHe "pHHHMaeT BH,IJ;
(U 2X2 + x 2u) dx - |
x 2(x du + u dx) = O. |
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I10CJIe y"porn;eHHil IIOJIyqHM: |
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u 2 dx - x du = 0 |
HJIH |
dx_du=O |
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X u2 |
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MHTerpHpY51 IIOCJIe,Il;Hee ypaBHeHHe, IIOJIyqHM In Ixl + ~ = C. BCIIOMHHM, qTO U = ~, HaXOMM 06rn;Hil HHTerpaJI HCXO,Il;HOrO ypaBHeHH5I: In Ixl + ~ = C.
OTMeTHM, qTO 3a,IJ;aHHOe ypaBHeHHe MO:lKHO 6bIJIO CHaqaJIa "pHBeCTH K
BHAY (2.2):
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2 + xy _ x 2 dy = 0 |
T.e. |
dy |
= |
y2 + xy |
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= |
(y)2 |
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HJIH |
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dx |
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+ x' |
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IIoJIarM y = ux, HaxO,Il;HM ,Il;aJIee yl = u1x + u H |
T.,Il;. (CM. 6)). |
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6) TIpe06pa3yeM ypaBHeHHe K |
BHAY (2.2): |
yl = (~)2 - |
~. TIOJIarM |
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y = ux, HaxO,Il;HM: yl = u1x + U. TIO,Il;CTaBHM 3HaqeHH5I y H |
yl B ,Il;aHHoe yPaB- |
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HeHHe: u1x + u = u 2 - |
u. TIpeo6Pa30BbIBM, IIOJIyqHM ypaBHeHHe C Pa3,Il;eJI5I- |
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IOrn;HMHC5I IIepeMeHHbIMH: ~: • x |
= u 2 - 2u. Pa3,Il;eJI5I5I IIepeMeHHble H HHTe- |
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rpHpy5l, HMeeM: f u 2~2u = f d: ' |
OTKYAa ~Inl u ~ 21 |
= In Ixl + ~In ICll, |
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T. e. |
Iu ~ 21 = |
ICl lx2 • |
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TIO,Il;CTaBJI5I5I |
u |
= ~, IIOJIyqaeM |
Iy ~2x I= |
ICl lx2 , |
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Y - 2x |
±C l X 2 , |
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Y - |
2x |
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Cx2 , r,Il;e |
C = ±C l • TeIIepb |
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T.e. |
-- y - = |
HJIH --y'- = |
Hail,IJ;eM |
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1+2 |
= C ·1,T. e. |
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3HaqeHHe IIOCT05lHHOil C, HCIIOJIb3Y51 HaqaJIbHoe YCJIOBHe: -1- |
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y-2x
C = 3. OTCIO,Il;a: -- y - = 3x2, T. e. y(3x2 -1) = -2x, oTKY,Il;a OKOHqaTeJIbHO:
2x
y = 1 _ 3x2 - qacTHoe pemeHHe 3a,IJ;aHHOrO ypaBHeHH5I.
B) TIpe06pa3yeM ypaBHeHHe K BHAY (2.2): yl-~+e~ = O. C,Il;eJIaB IIO,Il;CTa~
HOBKY ~ = U, T. e. y = ux, IIOJIyqHM u1x+u-u+eu = 0, HJIH d~+ ~ = O. MH-
e
TerpHpy5l, HMeeM:j e-u du = - f ~,T. e. -eu = -In lxi-In ICI, C "I- o. OT-
CIO,Il;a InlCxl = e-u , T.e. -u = InInlCxl, C"I- O. YqHTbIBM, qTO U =~, IIoJIyqaeM 06rn;ee pemeHHe 3a,IJ;aHHOrO ypaBHeHH5I y = -x In In ICxl, C "I- o. •
Pewum'b ypa6He'H.M: |
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xy + y2 |
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2.2.2. |
ydx + (x + y) dy = O. |
2.2.3. |
I |
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Y |
= 2X2 + xy . |
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2.2.4. |
xyl = y + xsin~, y(l) = ~. |
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J CooPHH" 3IIAII. DO BWCWeA ..""'..8TH.... 2 t<ypC |
65 |
2.2.5.IIpHBecTH ,n;Hci>ci>epeHIJ;HaJIbHOe ypaBHeHHe
(y + 2) dx - (2x + y + 6) dy = 0
a |
K O,n;HOPO,n;HOMY. |
IIoJIO>KHB x = U + 0, y = v + (3, nOJIyqaeM |
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(v + (3 + 2) du - (2u + 20 + v + (3 + 6) dv = 0, |
T. e. (v + «(3 + 2)) du - (2u + v + (20 + (3 + 6)) dv = O. IIo,n;6epeM 0 H (3 TaK,
qT06bI |
{ |
(3 + 2 = 0, |
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20 + (3 + 6 = o. |
PeIIIruI CHCTeMY, HaxO,n;HM, |
0 |
= -2, (3 = -2. Tor,n;a Hcxo,n;Hoe ypaBHeHHe |
npHHHMaeT BH,n; (2.1): v dv - |
(2u +v) dv = 0, T. e. HBJIHeTCH O,n;HOPO,n;HbIM, qTO |
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H Tpe60BaJIOcb. |
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• |
2.2.6.PeIIIHTb ypaBHeHHe, CBe,n;H ero K O,n;HOPO,n;HOMY:
(2x - 2) dy = (x + 2y - 3) dx.
2.2.7.HatiTH ypaBHeHHe KPHBOii, npoxo,n;HIIIeii qepe3 TOqKY A(I, 1), y KoTopoii no,n;KacaTeJIbHruI (CM. 3a.n;aqy 2.1.76) paBHa cyMMe KOOP,n;H- HaT TOqKH KacaHHH.
y
o
Puc. 6
a Ha pHC. 6 OTpe30K BC HBJIHeTCH no,n;KacaTeJIbHoii. KacaTeJIbHruI K HC-
KOMOii KPHBOii Y = |
f(x) npOBe,n;eHa B TOqKe M(x,y). TaK KaK no yCJIO- |
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BHIO BC = x + y, |
TO H3 npHMoyroJIbHOrO TpeyrOJIbHHKa MCB HaxO,n;HM: |
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tg 0 |
= -+y ,T. e. |
y' = -+y . PeIIIHM nOJIyqeHHOe |
O,n;HOPO,n;Hoe ,n;Hci>ci>e- |
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x |
y |
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x |
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y |
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y' = u'x + u, HMeeM |
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peHIJ;HaJIbHOe ypaBHeHHe. IIoJIarruI y = ux, |
oTKy,n;a |
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, |
+ u |
= |
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ux |
, |
x |
u |
0 |
du |
_u2 |
U X |
-+--, T. e. u |
= -1-- - u. |
TCIO,n;a |
-d x = |
-1--' HJIH |
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I+u |
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x |
UX |
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+u |
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x |
+u |
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dx |
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-- 2 - = |
-X· lIHTerpHpYH nOJIyqeHHOe ypaBHeHHe, HMeeM |
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u |
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In lui - ! = -In Ixl - |
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In ICI, |
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C¥- 0, T. e. ! = In ICxul HJIH |
W= In ICyl, C¥- O. IIo,n;cTaBJIHH x = 1, y = 1 |
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(no YCJIOBHIO KpHBaH npoxo,n;HT qepe3 TOqKY A(I, 1)), Haxo,n;HM KOHKpeTHoe
66
3Ha'IeHHeG: 1 = In IGI, G = ±e. TaKHM 06Pa30M, HCKOMOil: KPHBOil: 1IBJUleTC1I
JIHHH1I, onpe)J;eJI1IeMM ypaBHeHHeM x = yIn leyl. |
• |
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2.2.8. |
Hail:TH ceMeil:cTBo JIHHHiI:, KacaTeJIbHble K |
KOTOPbIM OTceKaIOT OT |
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OCH a6CIJ;HCC oTpe3KH, paBHble op)J;HHaTe TO'IKHKacaHH1I. |
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2.2.9. |
Hail:TH KPHBYIO, npoxoMIIJ;YIO 'Iepe3TO'IKYA(I, 1), Y KOTOPOil: pac- |
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CT01lHHe JII060il: KacaTeJIbHoil: OT Ha'IaJIaKoop)J;HHaT paBHo a6CIJ;HC- |
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ce TO'IKHKacaHH1I. |
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2.2.10. |
xy' = y + Jx 2 + y2. |
2.2.11. |
dy |
y |
x |
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dx = x |
y. |
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2.2.12. |
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2.2.13. |
xy' - |
y(lny -lnx) = O. |
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y = xy' - xex. |
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2.2.14. |
, |
= |
y + 2y'xY |
. |
2.2.15. |
ss' - |
2s + t = O. |
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y |
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x |
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2.2.16. |
x 2 + y2 = 2xyy'. |
2.2.17. |
Vfj(2vx-Vfj) dx+x dy = O. |
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2.2.18. |
, |
= |
x+y |
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2.2.19. |
y' cos ~ - |
~ cos ~ + 1 = O. |
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x _ y |
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2.2.20. |
xy' + xtg ~ = y. |
2.2.21. |
dy |
y |
+ lny -lnx) = O. |
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dx - |
x(1 |
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2.2.22. |
(3x2 - |
y2)y' = 2xy. |
2.2.23. |
y' - |
1 = e~ + ~, y(l) = O. |
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2.2.24. |
(2x3 y - |
y4) dx + (2 xy3 - |
x 4) dy = O. |
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2.2.25.xdy = (x + y) dx, y(l) = O.
2.2.26.y2 + x 2y' = xyy', y(l) = 1.
2.2.27.(y' -~) arctg~ = 1, Y G) = o.
=x+y
2.2.28.xy' - y (x + y) In - x - .
2.2.29. |
Y(X2 + y2) dx - x 3 dy = O. |
2.2.30. |
(x 2 + y2 + xy) dx - x 2 dy = O. |
2.2.31.x 2y' + xy - x 2 - y2 = 0, y(l) = O.
2.2.32.x 2 - 3y2 + 2xyy' = 0, y( -2) = 2.
2.2.33.y - xy' = 2(x + yy'), y(l) = O.
2.2.34.y' = ~ln~, y(l) = e.
2.2.35.Hail:TH KPHBYIO, npoxo)J;1IIIJ;YIO 'Iepe3TO'IKYA(I,O), eCJIH H3BeCT-
HO, 'ITOTpeyrOJIbHHK, 06pa30BaHHbliI: OCbIO op)J;HHaT, KacaTeJIbHoil: K KPHBOil: B npOH3BOJIbHoil: ee TO'IKe1'1 Pa,Il;HYC-BeKTopOM TO'IKHKaCaHH1I, paBH06e)J;peHHbliI:; OCHOBaHHeM ero 1IBJI1IeTC1I OTpe30K KacaTeJIbHoil: OT TO'IKHKaCaHH1I )J;O OCH op)J;HHaT.
2.2.36. Hail:TH KPHBYIO, npOXOMIIJ;YIO 'Iepe3TO'IKYA(I,2), )J;JI1I KOTOPOil: oTpe30K Ha OCH op)J;HHaT, oTceKaeMbliI: JII060il: KacaTeJIbHoil: K KPHBOO, paBeH a6CIJ;HCCe TO'IKHKaCaHH1I.
67
2.2.37. HaihH KPHBYIO, npOXO,l1,HUU'IO'Iepe3TO'IKYA(3, 0), eCJIH H3BecTHO,
'ITOyrJIOBoit K03<P<PHD:HeHT KacaTeJIbHoit paaeH
2.2.38. HaitTH ceMeitcTBo KPHBbIX, no,n:KacaTeJIbHruJ B mo6oit TO'IKeKOT0pbIX paaHa cpe,n:HeMY apH<pMeTH'IecKoMYKoop,n:HHaT TO'IKHKacaHHlI.
60nee CnO)l(Hbie 3aWI"'IM
2.2.39. PemHTb ypaBHeHHe, CBe,l1,H ero K O,n:HOPO,n:HOMY:
a ) |
I 3x - |
4y - |
2 |
6) I |
X + y - 2 |
y = 3x - |
4y - |
3 j |
Y = 3x - y - 2· |
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2.2.40. PemHTb ypaBHeHHe X 3 (yl - |
x) = y2. |
(C,n:eJIaTb 3aMeHY y = urn. |
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qHCJIO m no,n:06paTb TaK, 'IT06bInpHBecTH ypaBHeHHe K O,n:HOPO,n:-
HOMY·)
2.2.41. HaitTH 06In:Hit HHTerpaJI ,n:H<p<pepeHD:HaJIbHoro ypaBHeHHlI:
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I |
y |
<p(¥) |
6) xyl = 4V2x2 + y2 + Yj |
a) |
y = x + |
() j |
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<pI ¥ |
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0) |
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2 |
+ 10~ + 10. |
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3y' = |
Y2 |
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x
2.2.42.3aaa"ta 0 npOOlCe'ICmope. HaitTH <POPMY 3epKaJIa, OTpaJKaIOIn:erO Bce
JIy'IH,HCXO,l1,HIn:He H3 O,n:HOit TO'IKH,napaJIJIeJIbHO 3a,n:aHHoMY HanpaaJIeHHIO. (PaccMoTpeTb Ce'IeHHe3epKaJIa nJIOCKOCTbIO OXYj HCTO'IHHK JIy'Ieit (cBeTa) nOMeCTHTb B Ha'IaJIeKoop,n:HHaT, OCb Ox
HanpaaHTb napaJIJIeJIbHO OTpaJKeHHbIM JIY'IaM.)
2.2.43. IJPH KaKHX a H j3 ypaBHeHHe yl = 2x'" + 3yf3 npHBO,n:HTClI K o,n:-
HOPO,n:HOMY C nOMOIn:bIO 3aMeHbI y = urn? (CM. YKa3aHHe K 3a,n:a- |
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'Ie2.2.40.) |
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§ 3. JU1HEMHbiE YPABHEHLUI. YPABHEH~SI |
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6EPHYl1l1~ |
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yl +p(x)y = g(x), |
(3.1) |
r,ll;e p(x) H g(x) - HerrpepblBHhIe <PYHKIIHH (B '1acTHOCTH- |
rrocToHHHhIe), Ha3hIBa- |
eTClI J&tLHeilH"'-M ypaaHeHUe.M nepaozo nopJla7Ga. |
~ |
YpaBHeHHe |
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Xl +p(y)x = g(y) |
(3.2) |
HBJUleTCH J&uHeilH"'-M omHocumeJ&bHO x u Xl.
ECJIH g(x) == 0, TO ypaBHeHHe (3.1) rrpHHHMaeT BH,ll; yl +p(x)y = 0 H Ha3hIBaeTCH J&uHeilH"'-M OaHOpOaH"'-M. OHO HBJIlIeTCH ypaBHeHHeM C Pa3,ll;eJIHIOIIIHMHCH rrepeMeHHhIMH. B CJIY'Iaeg(x) 1- 0 ypaBHeHHe (3.1) Ha3hIBaeTCH J&uHeilH'W.M HeOaHOpOaH'W.M ypaaHeHUe.M.
68
PeIlIeHHe ypaBHeHHH (3.1) H~eTCH B BH,!I;e y = uv, r,!l;e u = u(x) H V = v(x) -
aeH3BeCTHhIe <PYHKIJ;HH OT X IIpH 3TOM O,!l;HY H3 3THX <PYHKIJ;HiI:
(aarrpHMep, v(x» MOJKHO BhI6paTb rrpOH3BOJIbHO (H3 c006paJKeHHiI: y,!I;06CTBa), TO-
r.ua BTOpaH Orrpe,!l;eJIHTCH H3 ypaBHeHHH (3.1). B 060HX CJIyqaHX OHH HaxO,!l;HTCH H3 YPaBHeHHiI: C pa3,!1;eJIHIO~HMHCH rrepeMeHHhIMH (CM. 3a,!1;aqy 2.3.1 a».
KpoMe Toro, ypaBHeHHe (3.1) MOJKHO peIlIHTb MeTO,!l;OM BapHau;HH rrpOH3BOJIbHOil: rrOCToHHHoil: (MemoiJ JIaepax:>tCa); B 3TOM CJIyqae em 06~ee peIlIeHHe H~eTCH
B BH,!I;e C(x)e-Jp(x)dx (CM. 3a,!1;aqy |
2.3.1 a». |
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~ YpaBHeHHe BH,!I;a |
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y' +p(x)y = g(x)yn, |
r,!l;e |
n E JR, n oF 0, n oF 1, |
a p(x) H g(x) - HerrpephIBHhIe <PYHKIJ;HH, Ha3hIBaeTCH ypa6xexueM Bepxy.ll.llu. ~
OHO rrpHBO,!l;HTCH K ,IlHHeil:HOMY ypaBHeHHIO C rrOMO~bIO rrO,!l;CTaHOBKH Z =Y- n+1. YpaBHeHHe BepHYJIJIH MOJKHO, He CBO,!l;H K JIHHeil:HOMY, rrpOHHTerpHpOBaTb C rrOMO-
~bIO rrO,!l;CTaHOBKH y = UV (T. e. MeTO,!l;OM BepHYJIJIH) HJIH rrpHMeHHB MeTO,!l; BapHaIJ;HH rrpOH3BOJIbHoil: rrOCToHHHoil: (MeTO,!l; JIarpaHJKa).
2.3.1. |
PeIIIHTb ,Il;H<p<pepeHIJ;HaJIbHble ypaBHeHH.H: |
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a) y' + tgx·Y = co~xj |
6) y' |
= ~j |
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0) xy' - 4y = x 2 y'Y. |
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x+y |
a |
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a) ,Il;aHHoe ypaBHeHHe HMeeT BH,Il; (3.1) H, CTaJIO 6bITb, .HBJI.HeTC.H JIHHefi- |
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HbIM. 3,Il;eCb p(x) = tgx, g(x) = co~x. PeIIIHM ypaBHeHHe ,Il;ByM.H cnoco6aMH. |
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MeTO,II. BepHYJIJIH |
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TIOJIaraeM y = uv, r,Il;e u = u(x), v = v(x) - |
HeKOTopble <PyHKIJ;HH OT x, |
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TOr,Il;a y' = u'v + uv' . ,Il;aHHOe ypaBHeHHe npHHHMaeT BH,Il;:
u |
I |
v + uv |
I |
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+ |
t |
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1 |
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g xuv |
= cos x ' |
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VIJIH |
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+ v tg x |
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1 |
(3.3) |
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I |
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( |
v |
I |
) |
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u |
v + u |
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= cos x . |
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ITO,Il;6epeM <PYHKIJ;HIO v = v(x) TaK, QTo6bI BblproKeHHe B cKo6Kax 6bIJIO paBHO HyJIIO, T. e. peIIIHM nepBoe ,Il;H<p<pepeHIJ;HaJIbHOe ypaBHeHHe C PM,Il;e-
JHlIOIIIHMHC.H nepeMeHHbIMH v' + v tg x = O. OTCIO,Il;a |
~~ + v tg x = 0, T. e. |
~ + tgxdx = 0, lnlvl-Inlcosxl = InIGI, G -:f. 0, |
OTKY,Il;a v = Gcosx, |
G ¥- o. TIOCKOJIbKY HaM ,Il;OCTaTOQHO KaKOro-HH6y,Il;b O,Il;HOrO HeHYJIeBoro pelIIeHH.H ypaBHeHH.H, TO B03bMeM v = cosx (nOJIO)KHJIH G = 1). TIO,Il;CTaBJI.H.H v = COSX B ypaBHeHHe (3.3), nOJIYQHM BTopoe ,Il;H<p<pepeHIJ;HaJIbHOe ypaBHeHIIe C pa3,Il;eJI.HIOIIIIIMIIC.H nepeMeHHbIMH, II3 KOToporo Hafi,Il;eM <PYHKIJ;IIIO u(x):
lId |
dx |
GT |
u cos x = cos x' T. e. |
u = --2-' H, CJIe,Il;OBaTeJIbHO u = tg x + . aKIIM |
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cos X |
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06PMOM, Y = uv = (tg x + G) cos X HJIII Y = G cos x +sin x - |
06IIIee peilieHIIe |
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IIcXOMoro ypaBHeHH.H.
69