Сборник задач по высшей математике 2 том
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X(P) = F(P) + <t>(P)X(P),
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a) x' - |
x = 1, x(O) = -1; |
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6) x" - |
2x' + 2x = 2t - 2, x(O) = x'(O) = 0; |
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B) XIII - |
x" = 4e2t , |
x(O) = 1, x'(O) = 2, x"(O) = 4. |
"* X(P). |
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x'(t): x'(t) "* pX(p) - x(O) = pX(p) + l.
Ih06paJKeHHeM <PyHKIJ;HH 1 jlBJIjleTCjI ~. TaKHM 06pa30M, npHMeHjIjI npe-
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CJIe,n:oBaTeJIbHO, f(t) = -l.
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x'(t) "* pX(P) - |
x(O) = pX(P), |
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x"(t) "* p2 X(P) - px(O) - |
x' (0) |
= p2 X(p). |
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p2 |
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2(1- p) X(p) = p2(p2 _ 2p+ 2)
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2(1 - p) |
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Ii - |
---::----''---- = |
p2 |
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p2 _ 2p + 2 |
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CJIe.n;OBaTeJIhHO, X (p) |
t;- t |
- sin t . et . |
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0) llYCTh x(t) -;-t |
X(p). Tor.n;a |
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x"(t) = p2 X(p) - px(O) - x'(O) = p2 X(P) - |
P - |
2, |
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xlll(t) =p3 X(P) - |
p2x(0) - |
px'(O) - x" (0) |
=p3 X(P) - p2 - 2p - |
4. |
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Iho6pIDKeHHeM rrpaBoit 'IacTHypaBHeHHH 6y.n;eT <PYHKIIHit ~2' OTcIO.n;a |
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nOJIY'IaeMorrepaTopHoe ypaBHeHHe |
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p- |
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p3 X (p) _ p2 _ 2p _ 4 _ p2 X (p) + P + 2 = _4_. |
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p-2 |
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PelIlHB ero OTHOCHTeJIhHO <PYHKIIHH X(P), rrOJIY'IHMX(p) = |
~2 H, CJIe.n;o- |
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BaTeJIhHO, x(t) = e2t • |
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p- |
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8.3.2. |
HaitTH o6ru;ee pelIleHHe ypaBHeHHH x" - 2x' + x = et . |
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rrpOH3BOJIhHhIe Ha'IaJIhHhIe YCJIOBHH 3a.n;a'IH KOIIlH. |
llYCTh |
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x(O) = Cl H x'(O) = C2. llYCTh Terreph x(t) -;-t |
X(p). Tor.n;a |
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x'(t) -;-t pX(p) - Cl |
H x"(t) -;-t |
p2 X(P) - |
CIP - |
C2. |
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TaK KaK et -;-t |
~1' TO COOTBeTCTBYIOru;ee orrepaTopHoe ypaBHeHHe HMeeT |
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p- |
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p2 X(P) - |
CIP - |
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C2 - |
2pX(P) + 2Cl + X(P) = --1' |
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p- |
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Haxo.n;HM OTCIO.n;a X (P): |
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X(P - |
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Cpeacmea.MU onepamopHoeo UC"I.UC.II.eH'Wl pew,um'b .ll.UHei1.H'bte (OaHOpOaH'bte U HeOaHOpOaH'bte) au!fj!fjepeH1l,Ua.ll.'bH'bte ypaeHeH'Wl (ec'/Oay x = x(t»):
8.3.3. |
x' + 3x = 0, |
x(O) = 2. |
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8.3.4. |
x' - 4x = 1 - |
4t, |
x(O) = l. |
8.3.5. |
x' + x = 2 cos t, |
x(O) = O. |
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8.3.6.x" + 4x' - 5x = 0, x(O) = 3, x'(O) = -3.
8.3.7. x" - 6x' + 9x = 0, x(O) = 1, x'(O) = 2.
511
8.3.8.x" + 4x = 0, x(O) = 1, x'(O) = 6.
8.3.9.x" + x' - 2x = 1, x(O) = 0, x'(O) = -2.
8.3.10. |
x" - |
3x' + lOx = 9sint - 3cost, x(O) = 0, x'(O) = -2. |
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8.3.11. |
x" - |
4x' + 4x |
= 4t, |
x(O) = 4, x'(O) = 7. |
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8.3.12. |
x" + 2x' + x = t + 2, |
x(O) = 0, x'(O) = 2. |
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8.3.13. |
x" - |
2x' + 5x |
= 1 - |
t, |
x(O) = x'(O) = O. |
8.3.14. |
x" - |
2x' + 2x |
= 1, |
x(O) = x'(O) = O. |
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8.3.15.x" - x' = et , x(O) = x'(O) = 4.
8.3.16.x" + x = 1, x(O) = -1, x'(O) = O.
8.3.17.x" - x = sint, x(O) = -1, x'(O) = O.
8.3.18.x" - x = 2sht, x(O) = 0, x'(O) = l.
8.3.19. |
x" + 2x' |
+ x = t, x(O) = x'(O) = O. |
8.3.20. |
x" + 2x' |
+ lOx = sin3t + 6 cos 3t, x(O) = x'(O) = l. |
8.3.21.XIII + x" = 0, x(O) = 2, x'(O) = X"(O) = l.
8.3.22.XIII + x' = 0, x(O) = 1, x'(O) = 2, X"(O) = 3.
8.3.23.XIII - 3x" + 3x' - x = 0, x(O) = 1, x'(O) = X"(O) = O.
8.3.24.xIV - X" = 0, X(O) = X'(O) = X"(O) = 0, XIll(O) = 2.
8.3.25.XIII_X" = -2et sint, X(O) = 1, x'(O) = 1, X"(O) = 2.
8.3.26. |
XIII - x' = cos t, x(O) |
= x' (0) = x" (0) = O. |
8.3.27. |
XIII + 2X" - 3x' = 4et , |
x(O) = x' (0) = x" (0) = O. |
8.3.28.xIV - X = t 3 , X(O) = 3, x'(O) = 1, X"(O) = 3, XIll(O) = l.
8.3.29. |
xIV - 2X" + X = 4tsint - |
8cost, |
X(O) = 1, x'(O) = 2, X"(O) = 1, |
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XIII (0) = 4. |
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8.3.30. |
xIV - X" = 1, x(O) = x' (0) = x" (0) = XIII (0) = O. |
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Hai1mu o6w,ue pewe'HUSI aug)(pepe'Hqua.l!'b'H'btx ypa6'He'Hui1: |
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8.3.31. |
X"_X = O. |
8.3.32. |
x" + 2x' + x = l. |
8.3.33. |
x" + x' - 2x = 3et . |
8.3.34. |
x" - 2x' + 5x = 5t2 + t. |
8.3.35.XIII + x' = 2.
8.3.36.PeIIIHTh 3a,llW-1Y KOiliH
{x" + x = f(t), x(O) = x'(O) = 0,
a |
r,ll;e f(t) |
3a,ll;aHa rpruPW-IeCKH (pHC. 116). |
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IIycTh x(t) |
-;-t |
X(P). Tor,ll;a x"(t) -;-t p2 X(P). Hafi,ll;eM H306proKeHHe |
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<PYHKIJ;HH f(t). |
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IIpHMeHHM BTOpofi BapHaHT. <l>YHKIJ;HH f(t) MO:>KeT 6hITh 3aIIHcaHa B BH,ll;e:
f(t) = x(t) - |
2x(t - 1) + x(t - 2). |
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Tor,ll;a |
1 |
2e-P |
e-2p |
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f(t) -;-t |
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p - |
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512
f(t)
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-1 -----~ |
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Puc. 116 |
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Puc. |
117 |
3amuneM Teneph onepaTopHoe ypaBHeHHe: |
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p2 X(P) + X(p) = 1 - 2e-P + e-2P • |
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p |
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Haxo.n;HM H3 Hero HeH3BeCTHoe H306prolwHHe X (P): |
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1 - |
2e-P + e- 2p |
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X(p) = |
p(P2 + 1) . |
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MeTo.n;OM Heonpe.n;eJIeHHhlX K09<pqmIl;HeHToB (HJIH )Ke HCrrOJIh3YH TeopeMY |
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p(p21+ 1) |
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p- p2+1' |
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CJIe.n;OBaTeJIhHO, |
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X(P) = (1 -_1_) (1 - 2e-P+ e- 2P ) |
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p p2 + |
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Ern;e pro HCrrOJIh3YH TeopeMY 3arra3.n;hlBaHHH, HaxO.n;HM HCKOMhlft OpHrHHaJI f(t) H306pa)KeHHH X(p):
f(t) = (1- cost)X(t) - 2(1- cos(t -1))x(t -1) + (1- cos(t - 2))X(t - 2) . •
Ha-tJ.mu "I.acm'H'bte pewe'HUSI aurjirjiepe'Hv,ua.!l(b'H'btX ypa6'He'Hu-tJ.;
8.3.37. |
x' + x = X(t - |
1), |
x(O) = O. |
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8.3.38. |
x" - x' = f(t), |
x(O) = |
x'(O), <PYHKIl;HH |
f(t) |
3a.n;aHa rpa<PHKoM |
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(pHC. 117). |
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= 1, x'(O) = 0, |
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8.3.39. |
x" + x' = f(t), |
x(O) |
<PYHKIl;HH f(t) 3a.n;aHa |
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rpa<PHKoM (pHC. 118). |
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8.3.40. |
x" +4x = f(t), |
x(O) = x'(O) = 0, <PYHKIl;HH f(t) 3a.n;aHa rPa<PHKoM |
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(pHC. 119). |
x(O) = x'(O) = 0, <PYHKIl;HH f(t) |
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8.3.41. |
x" - x = f(t), |
3a.n;aHa rpa<pHKoM |
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(pHC. 120). |
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17 CooPHH" _~ no &woweR MBTCMIlTHKC. 2 ocypc |
513 |
f(t)
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f(t) |
1 |
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1 |
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Puc. 118 |
Puc. 119 |
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f(t) |
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1 |
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Puc. 120 |
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8.3.42. HaihH peIIIeHHe 3a,n;a<JH KOIIIH |
tx" + tx' - X |
0, x(O) = 0, |
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x'(O) = l. |
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IIycTb x(t) ~ X(P). Tor,n;a |
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x'(t) ~ pX (P) ,
x"(t) = p2 X(P) - x'(O) = p2 X(p) -l.
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tx'(t) ~ - ~(PX(P)) = -X(p) - pX' (P) , tx"(t) ~ - ~ (p2 X(p) - 1) = -2pX(p) - p2 X'(p).
CJ1e,n;OBaTeJ1bHO, onepaTopHoe ypaBHeHHe npHMeT BH,n;:
-2pX(P) - p2 X'(P) - X(p) - pX'(P) - X(p) = 0,
HJ1H
(P2 + p)X'(p) + 2(p + I)X(P) = 0,
oTKy,n;a
pX'(p) + 2X(P) = O.
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dX |
2dp |
C |
X |
P' |
oTKy,n;a In X = -2Inp + In C, T.e. X(p) = 2". |
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514
OpHnma.nOM ,11,JUI 9TOii <PYHKIl,HH CJIY:lKHT <PYHKIl,H5I |
X(t) = C· t. MCrrOJIb- |
3y5I Haqa.nbHOe YCJIOBHe X'(O) = 1, HaXO,l]HM C = 1. |
OKOHqaTeJIbHO HMeeM |
x(t) = t. |
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Hat1.mu "I.acm'H'bte peme'H'IJ.R aurjirjiepe'Hv,ua.!l,'b'H'btX ypa6'He'Hut1. cpeaCm6a.MU onepav,uo'H'Hoeo uC"I.UC.I!e'H'!.tJI:
8.3.43.tx" - 3x' = 0, x(O) = 2, x' (0) = O.
8.3.44.tx" + 2x' = 0, x(O) = 1, x'(O) = O.
8.3.45. |
tx" + tx' + x = 0, x(O) = 0, x'(O) = 1. |
8.3.46. |
PelllHTb CHCTeMY JIHHeiiHbIX ypaBHeHHii |
X' +y = 2et , |
y(O) = 1. |
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{ |
x(O) = |
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y' +x = 2et , |
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a lIYCTb x(t) -;-t X(P) H y(t) -;-t |
Y(p). YqHTbIBM, qTO e t -;-t p ~ l' rrOJIY- |
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qaeM orrepaTopHYIO (HJIH H306pruKa1OIIl,yIO) CHCTeMY JIHHeiiHbIX OTHOCHTeJIbHO <PYHKIl,Hii X (p) H Y (p) ypaBHeHHii
PX(P) - 1 + Y(p) = p':' l' |
{:} { |
PX(P) + Y(p) = P ~ ~, |
{ |
P 1 |
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pY(p) -1 +X(P) = ~1' |
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X(p) + pY(p) = P + 1. |
p- |
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p- |
PelliM 9TY cHcTeMY, rrOJIyqHM X(p) = Y(p) = ~1. lIo Ta6JIHIl,e H306pruKe- |
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HHii HaXO,l]HM Terrepb x(t) = e t |
p- |
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H y(t) = e t . |
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Pemumb CUcme.M'bt ypa6'He'Hut1.: |
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8.3.47. |
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X' + y = 0, |
= 1, y(O) = -1. |
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{ |
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x(O) |
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y' +x =0, |
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8.3.48. |
{ |
X' - |
3x - 4y = 0, |
x(O) = y(O) = 1. |
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y' - |
4x + 3y = 0, |
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8.3.49. |
{ |
X' + X - y = 2, |
X (0) = 0, y (0) =-1. |
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y' +x +y = 2t, |
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X' + y + z = 0, |
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8.3.50. |
{ |
y' + X + z = 0, x(O) = 1, y(O) = 0, z(O) = -1. |
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z' +x + y = 0, |
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X' + y = t, |
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8.3.51. |
{ |
y' + z = t 2 + 1, |
x(O) = 1, y(O) = z(O) = O. |
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z' +x = 2t+ 1, |
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8.3.52. |
{ x" - |
2y' - x = 0, |
x(O) = 0, x'(O) = y(O) = 1. |
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y' + x' - X - Y = et , |
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515
8.3.53. { 2X" - |
x' + 9x |
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y" - y' - |
3y |
= 0, x(O) x'(O) = 1, y(O) = |
2x" + x' + 7x |
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y" + y' - |
5y |
= 0, |
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=y'(O) |
= O. |
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8.3.54.PemHTh HHTerpa.nhHhle ypaBHeHHH:
t |
6) x(t) - /(t -T)X(T) dT = sint. |
a) / et-rx(T) dT = tj |
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0 |
Q a) MHTerpa.n, CTOHID;Hit B JIeBoit qacTH ypaBHeHHH, npe.n;CTaBJIHeT co6oit CBepTKY <PYHKII;Hit et H x(t). IIyCTh x(t) -!+ X(P). Tor.n;a no TeopeMe 0 CBepTKe nOJIyqHM H306proKeHHe HHTerpa.na
t
/ et- r X(T) dT = et *x(t) -!+ P ~ 1X(p). o
CocTaBHM Teneph onepaTopHoe ypaBHeHHe:
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oTKy.n;a |
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X(P) = -- = - - -. |
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CJIe.n;OBaTeJIhHO, x(t) |
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6) IIyCTh x(t) -!+ |
X(p). IIo Ta6JIHII;e H306proKeHHit HaxO.n;HM |
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IIo TeopeMe 0 CBepTKe nOJIyqHM H306proKeHHe HHTerpa.na: |
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/(t -T)X(T) dT = h |
x(t) -!+ |
\X(p). |
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COCTaBJIHeM onepaTopHoe ypaBHeHHe |
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X(P) - |
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PemM ero OTHOCHTeJIhHO <PyHKII;HH X (P), HaxO.n;HM |
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Pewumb UHme1!pa.tlbH'bte ypa6HeH'U.R: |
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8.3.55. |
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et- rx(T) dT = sin t. |
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8.3.56. |
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COST' x(t - T) dT = sin t. |
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8.3.57./cos(t-r)x(r)dr=t2.
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8.3.58./e2(t-U)x(u)du = t2et . o
8.3.59. |
/(t -r)x(r) dr - x(t) = - cost. |
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8.3.60. |
x(t) = |
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8.3.61. |
/(t -r)2x(r) dr - |
2x(t) + 2et = O. |
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sin(t - u)] x(u) du = t. |
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8.3.62. |
x(t) - |
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8.3.63. |
/(1- 2(t - r))x(r) dr - x(t) = 2 (1 + t - et ). |
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8.3.64. |
x(t) = 1 + ~ /(t -U)3 x (U) duo |
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8.3.65. |
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sh(t - r)x(r) dr + x(t) = t. |
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8.3.66. |
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x(u) du + /(t -u)x(u) du + x(t) = t. |
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Pewumb i}u!fj!fjepeH'qua.n,bHbte ypa6HeHUSI cpei}cm6a.MU onepa'qUOHHO(!O UC"tU- C.n,eHUSI:
8.3.67.x" + 3x' = et , x(O) = 0, x'(O) =-1.
8.3.68.x" - 4x' + x = 1 - 2et , x(O) = 2, x' (0) = 1.
8.3.69. x" + 2x' + x = t2, x(O) = 1, x' (0) = O.
8.3.70.x" + x = cost, x(O) = -1, x'(O) = 1.
8.3.71. |
XIII - |
x' + 3x = 12 +3sint - 2 cost, x(O) = 4, x'(O) = 1, x" (0) = O. |
8.3.72. |
x" + x = 1, x(O) = -1, x' (0) = O. |
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8.3.73. |
x" - |
x' = tet , x(O) = x' (0) = O. |
8.3.74. |
XIII - |
2x" + x' = 4, x(O) = 1, x'(O) = 2, x"(O) = -2. |
8.3.75.XIII + x' = e2t , x(O) = x' (0) = x" (0) = O.
8.3.76.xIV - x = 1, x(O) = 0, x'(O) = 3, x"(O) = -1, x"'(O) = 1.
8.3.77.XIII +x = 1, x(O) = x'(O) = x"(O) = O.
517
f(t)
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Puc. |
121 |
8.3.78. x' - x = f(t), x(O) = 0, |
<PYHKIl,HH f(t) 3a,L1;aHa rpa<PHKoM (CM. |
pHC. 121). |
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Ha11.mu o6w,ue peme'H.USI aurfirfiepe'H.tJ.,ua.!l'b'H.'btx ypa6'H.e'H.u11. cpeaCm6a.MU onepa.- tJ.,UO'H.'H.OZO uC"I.UC.!Ie'H.USI:
8.3.79. |
x" - 4x' = t. |
8.3.80. |
x" + 2x' + x = t2 + 5t + 4. |
8.3.81. |
x" + x = 2 cos t. |
8.3.82. |
XIII - x" = et . |
8.3.83.xIV - 8X" + 16x = COS t.
Pemum'b U'H.mezpa.!l'b'H.'bte U u'H.mezpo-aurfirfiepe'H.tJ.,ua.!l'b'H.'bte ypa6'H.e'H.USI:
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8.3.84. |
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ch(t - |
u)x(u) du = sh t. |
8.3.85. f ch U· x(t - u) du = t. |
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8.3.86. |
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x(r)x(t - r) dr = ~ sint - |
~tcost. |
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8.3.87. |
x(t) - t = ~ f(t - r)2 x (r) dr. |
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8.3.88. |
u)x(u) du - x(t) + 1 + t = O. |
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8.3.89.2x(t)-2= fSin2(t-r)x(r)dr.
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8.3.90. |
x'(t) + f |
x(r) dr = 1, x(O) = O. |
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8.3.91. |
x'(t) + f(t - r)x(r) dr = 1 + t, x(O) = O. |
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Pemum'b cucme.M'bt |
aurfirfiepe'H.tJ.,ua.!l'b'H.'btx ypa6'H.e'H.u11.: |
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8.3.92. |
X' |
- 2x + y = 3 - 2t, |
y(O) = 2. |
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+ x + 2y = 4 + t, |
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518
8.3.93. |
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X' + X - y = sint, |
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x(O) = 0, y(O) = l. |
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8.3.94. |
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2y + 2x = 2t2 , |
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8.3.95. |
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x(O) = 1, y(O) = 2, z(O) = l. |
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X + y = 1, |
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x(O) = y(O) = 1, z(O) = O. |
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8.3.96. |
{ |
y' |
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X' + y + z = 2et |
+ 3, |
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8.3.97. |
{ |
y' + X + z = 2et |
+ 2, x(O) = 1, y(O) = 3, z(O) = l. |
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Zl + X + y = 2et |
+1, |
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KOHTPOJlbHAH PA60TA |
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Bapll1aHT 1 |
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1. HafiTH H306pIDKeHHH CJIe~IOm;HX 0pHrHHaJIOB: |
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f(t) = tcos2 t; |
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3t |
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6) f(t) = +. |
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HafiTH H306pIDKeHHe nepHO,Il;H'-leCKOfi<PYHKUHH C nepHO,Il;OM T = 1, 3a,Il;aH- |
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Hofi Ha OTpe3Ke [0,1] paBeHCTBOM f(t) = 1 - |
t. |
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3. HafiTH OpHrHHaJIbI no CJIe~IOm;HM H306pIDKeHlfHM: |
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a) |
F(P) - |
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P +1 . |
6) F(P) _ |
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p3 + 4p2 + 5p' |
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4. |
PeIIIHTb ,Il;H<p<pepeHUHaJIbHOe ypaBHeHHe |
x" + x' - |
2x = et , |
x(O) = 1, |
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X'(O) = O. |
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2 |
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5. |
PeIIIHTb HHTerpaJIbHOe ypaBHeHHe x(t) = |
t2 + f(t - r)e-(r-t)x(r) dr. |
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Bapll1aHT 2 |
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1. HafiTH H306pIDKeHHH CJIe~IOm;HX 0pHrHHaJIOB: |
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a) |
f(t) = t2 sin2t; |
6) f(t) = cos3t - |
cost. |
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t |
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2. HafiTH CBepTKY H ee H306pIDKeHHe ,Il;JIH <PYHKUHfi f(t) |
= t2 , g(t) |
= sin t. |
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519