Сборник задач по высшей математике 2 том
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8.2.2. |
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8.2.3. |
F(P) - |
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F(p) = P |
+ p3 |
+ P + 1· |
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(P + 3)5 |
(p _ 4)4 . |
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8.2.4. |
F(P) = 3p + 1. |
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8.2.5. |
F(p) = 5p- 3. |
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p2 +9 |
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p2 _ 4 |
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8.2.6. |
F()- |
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8.2.7. |
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3 -4p |
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2p+ 2· |
F(p) = p2 +4p+8 |
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8.2.8. |
F(P) = 3e- |
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8.2.9. |
F(p) = |
e-P + e-2p |
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(P _ 2)2 . |
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p+3 |
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(e- P - 3e-4p )p |
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8.2.10. |
F( |
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8.2.11. |
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F(P) = |
p2 + lOp + 29 . |
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8.2.12.HaiiTH OpHrHHaJIbI CJIe.rr.yIOIII.HX H306pIDKeHHii:
a) F(P) - |
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p(p+l)' |
p - p(P-l)(p2+1)' |
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B) F(p) = (P _ 1)3; |
r) F(p) = (p2 + 1)2 |
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p(P + 1) |
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B: HMeeM paBeHcTBo 2 = A(P + 1) + Bp, KOTopoe CnpaBe,ll;JIHBO ,ll;JIj{ Bcex 3HaqeHHii p. nOJIarM p = -1 H P = 0, nOJIyqaeM ,ll;Ba COOTHOIIIeHHj{ 2 = -B
H 2 = A, H3 KOTOPbIX HaxO.n;HM A = 2, B = -2. OTcIO.n;a
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= 4 + -..lL + Cp + D |
p(p - 1)(p2 + 1) |
P P - 1 ~ + 1 . |
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qHCJIHTeJIH o6eHx ,lJ,po6efi, nOJIyqaeM paBeHcTBo:
1 = A(P - 1)(p2 + 1) + Bp(p2 + 1) + (Cp + D)p(P - 1).
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p=O |
1 =-A |
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p=1 |
1 = 2B |
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p= -1 |
1 |
= -4A - 2B - 2C + 2D |
p=2 |
1 |
= 5A + lOB + 4C + 2D |
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A = -1, |
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{ 2B = 1, |
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A+B+C=O, |
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-4A - 2B - |
2C + 2D = 1, |
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1 |
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OTKY,lJ,a A = -1, B = 2'C = 2'D = |
-2·TaKHM o6pa30M, |
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1 |
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= _1 + 1. _1_ + 1 p - 1 = |
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p(p - 1) (p2 + 1) |
P |
2 p - |
1 |
2 p2 + 1 |
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= _1 + 1. _1_ + 1. _P_ _ 1. _1_. |
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p |
2 p - l |
2 |
p2+1 |
2 |
p2+1 |
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---,------''-:::------:- +;- |
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+ -2 |
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p(p - |
1)(p2 |
+ 1) . |
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B) 3anHrneM F(p) B BH,lJ,e: |
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p |
(p - |
1) + 1 |
1 |
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(p-3)3 = |
(p_l)3 |
(P_l)2 + (P-l)3 |
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p |
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p2 -1 |
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p2 + 1 ' |
p2 + l' |
(P2 + 1)2' |
(p2 + 1)2' |
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t sin t, |
t cos t |
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Bp |
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+ |
D(p2 - |
1) |
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---::-=-~ = -- |
+ -- + |
(P2 + 1)2 |
(P2 + |
1)2 |
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(p2 |
+ 1)2 |
p2 |
+ 1 |
p2 + |
1 |
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llCnOJIb3YH npHeMbI, H3JI0:lKeHHbIe BbIIIIe, HaxO.n;HM
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1 |
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A = 2"' B |
= C = 0, D = -2". |
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CJIe.n;OBaTeJIbHO, |
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IIo 3aaaHH'bt.M u306po:JlCeHUSI.M Hal1mu OpUeUHaA'bt: |
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8.2.13. |
F(P) = p~2. |
8.2.14. |
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3p-4 |
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F(p) |
= (p _ 1)(P _ 2) |
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P |
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8.2.15. |
F(p) = 2p + 1. |
8.2.16. |
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2-p |
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p2 _p |
F(p) = (p-1)2. |
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8.2.17. |
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2p3 + p2 + 2p - |
1 |
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6p2 -p-6 |
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F(P) = |
p4 -1 |
8.2.18. |
F(P) |
= p3_ p2_6p |
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p3+ p2_1 |
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3p2 - 2p+ 5 |
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8.2.19. |
F(p) = |
p4 _p3 |
8.2.20. |
F(p) = p3 _ 2p2 + 5p |
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8.2.21. |
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p2 |
8.2.22. |
F(P) |
= |
p3 +2r +4p+2 |
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F(P) = (p + 3)4 . |
p4 + 5p2 + 4 |
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8.2.23. |
F(P) - |
1 |
8.2.24. |
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p2 _ 2p-1 |
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F(P) |
= p3 _ 3p2 + 3p - |
1 |
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(p-1)2(P+2) |
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8.2.25. |
F(P) = |
p2_p+1 |
8.2.26. |
F(p) = |
2p3 - p2 + 4p - |
4 |
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(p2 _ 1)2 . |
p4 _ 8p2 + 16 |
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8.2.27. |
F _ |
(P+1)2 |
8.2.28. |
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2p3 - 2p2 - 2 |
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(p) - |
(p2 + 1)2 |
F(P) = (p2 _ 2p + 2)2· |
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8.2.29. |
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(2p2 - 5)e-3p |
8.2.30. |
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p2e-P |
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F(P) = p4 _ 5p2 + 4 |
F(P) |
= (P + 2)5· |
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8.2.31.Hail:TH cBepTKY <PYHKll;Hil: f(t) H g(t)H ee H306paJKeHHe:
a) f(t) = et , g(t) = tj |
6) f(t) = cost, g(t) = e2t. |
Q a) IIep6'bt11 cnoco6. |
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Hail:.n;eM no Ta6JIHll;e H306paJKeHHH <PYHKll;Hil::
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e *t * p -1 r |
= p2(P -1)· |
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et *t -!-t et - 1 - t.
Bmopofi. cnoco6.
BbI'IHCJIHMCBepTKY <PYHKU;Hit, nOJIb3YJlCb onpe,n:eJIeHHeM:
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et *t = !t eT(t - |
r) dr. |
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HHTerpHPyeM no 'IacTJlM: |
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t |
t |
t |
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!eT(t - |
r) dr = !(t - r) deT = (t - r)eT10 - |
!eTd(t - r) = |
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0 |
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t |
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=-t+ !eTdr=-t+eTlo=-t+et-l. o
CJIe,n:OBaTeJIbHO, et *t = et - t - 1. Tenepb no Ta6JIHu;e H306paJKeHHit HaxO,n:HM
M306paJKeHHe cBepTKH:
et *t -!-t _1__ l _ 1 = |
1 . |
p - 1 p2 P |
r(P - 1) |
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HMe CBepTKH. HMeeM: f(t) = cost -!-t |
+- H g(t) = e2t -!-t ~2' Tor,n:a |
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p + 1 |
p- |
f(t) * (t) -!-t |
_p_. _1_ - |
P |
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p2 + 1 |
p - 2 - |
(P2 + 1)(P - 2) |
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P |
= -"5P +"5 + -.L = _~_P_ + 1_1_ + ~_1_. |
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(P2+1)(P-2) |
p2+1 |
p-2 5 p2 +1 5 p2 +1 |
5p-2 |
no Ta6JIHu;e H306paJKeHHit HaxO,n:HM Tenepb caMY CBepTKY: |
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cos t *e2t |
= - ~cos t + isin t + ~e2t. |
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Hafi.mu C6epm'IGY giY'H:K:u,ufi., UCnO.lI:b3YJI onpeoe.n,eHue:
8.2.32.f(t) = 1, g(t) = 2. 8.2.33. f(t) = 1, g(t) = t.
8.2.34. |
f(t) |
= sint, g(t) |
= cost. |
8.2.35. |
f(t) = et , g(t) = e2t . |
8.2.36. |
f(t) |
= cost, g(t) |
= t. |
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503
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8.2.37.f(t) = sin t, g(t) = sin 2t. 8.2.38. f(t) = e3t , g(t) = t2.
8.2.39. f(t) = cos t, g(t) = t3. |
8.2.40. f(t) = sh t, g(t) = ch t. |
B cAeayto14Ux 3aaa"tax Ha11,mu C6epm'K:y !PYH'K:'qu11, f(t) Ug(t) U ee u306pa:HCeHue:
8.2.41.f(t) = 1, g(t) = et. 8.2.42. f(t) = e3t , g(t) = e5t .
8.2.43.f(t) = et , g(t) = sin t. 8.2.44. f(t) = t2, g(t) = cos 2t.
8.2.45. |
f(t) = t, g(t) = tet . |
8.2.46. |
f(t) = sin t, g(t) = sin t. |
8.2.47.f(t) = t, g(t) = t sin t.
8.2.48.Hailnl cBepTKY <PYHKIJ;Hil X(t) H X(t - 1).
8.2.49.ITYCTb f(t) = etX(t). HailTH CBepTKY <PYHKIJ;Hil f(t - 1) H f(t - 2).
8.2.50.ITOJIb3YJlCb TeopeMoil 0 cBepTKe, HailTH OpHrHHaJI H306pIDKeHHJI
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F(P) - |
p |
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(P2 + 1)2 |
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Q ITpe,n:CTaBHM F(P) B BH,n:e npoH3Be,n:eHHJI |
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F(P) - |
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P_ |
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-p2+1p2+t" |
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<l>YHKIJ;HH ~ H -I-- JlBJIJlIOTCJI H306pIDKeHHJlMH <PYHKIJ;Hil sin t H cos t |
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P +1 |
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COOTBeTCTBeHHO. ITo TeopeMe 0 cBepTKe HMeeM: |
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2 |
p |
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t;- sin t *cos t. |
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(P |
+ |
1) |
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OCTaJIOCb HailTH cBepTKY <PYHKIJ;Hil sin t |
H cos t: |
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t |
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t |
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sin h |
cost = Jsin rcos(t - |
r) dr = J~[sint + sin(2r - |
t)] dr = |
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o |
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0 |
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= ~ (rsint- ~cos(2r-t)) I: = ~ (tsint- ~cost+ ~cost) = ~tsint. |
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TaKHM 06Pa30M, |
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(P2 + 1)2 t;- |
2 sm . |
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2 MO:lKHO 6bIJIO HailTH HHa'le,BOCnOJIb- |
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nOA'b3YSlC'b meopeMo11, 0 C6epm'K:e: |
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8.2.51. |
F(P) - |
3 |
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8.2.52. |
F( |
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r |
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(P2 + 4)2 |
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(p2 + 1)2 |
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8.2.53. |
F( |
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1 |
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8.2.54. |
F( |
p |
) - |
1 |
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p - |
(p2 _ 2p + 2)2 |
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(P2 + 1)(P2 + 9) |
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504
8.2.55.BbI'iHCJIHTbHHTerpa.nbI:
t
a) I et- x sin x dx;
o
t
B) Ie-x cosxdx. o
o a) llHTerpa.n
t
I et- x sin x dx o
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1 |
1 |
1( |
1 |
1 - |
PI) |
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F(P) = p2 + 1 P - |
1 = 2 |
P - |
p2 + 1 - p2 |
+ 1 . |
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OPHrHHa.nOM 3Toro H306paJKeHHH CJIY)KHT <l>YHK~HH |
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f(t) = ~ (et - cost - sint). |
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CJIe,Il;OBaTeJIbHO, |
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I et- x sinxdx = ~ (e t |
- cost - sint) . |
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6) PacCMOTPHM <l>YHK~IO |
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t |
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OHa rrpe,Il;CTaBJIHeT co60it CBepTKY <l>YHK~Hit t H et • llMeeM |
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t *et -# 1 1 |
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ITo Ta6JIH~e HaxO,Il;HM: |
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TaKHM o6pa30M, f(t) = t *et |
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le1-Xxdx = f(l), |
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B) npeo6pa3yeM HHTerpa.n: |
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I e- x cosxdx =e-t |
• et I e- x cosxdx =e-t |
I et- x cosxdx. |
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000
505
ft et- x cos x dx = cos t *et . o
Era H306paJKeHHeM 6Y)J,eT <PYHKll;HJI ~ . ~1. Tor)J,a no TeopeMe CMe- |
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ru;eHHJI H306paJKeHHJI HMeeM: |
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OpHrHHaJIOM nOCJIe)J,Hera BhlpaJKeHHJI CJI)')KHT<PYHKll;HJI |
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CJIe)J,OBaTeJIbHO, |
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e-t ft et- x cos x dx = ~ (1 + e-t sin t - |
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oTKY)J,a OKOH'IaTeJIbHOnOJIY'IaeM: |
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ft e- x cosxdx = ~ (1 + e-t sin t - e-t cost) . |
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B'b/,"I,UC.I/,Umb |
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8.2.56. |
fsin r . sin(t - r) dr. |
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j{t - X)2 cos 2x dx. |
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f sin x . e"2- x dx. |
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x) sin(7r - |
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a) |
F(p) = |
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6) F(P) = (p2 _ 1)2; |
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F(p) = |
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Q a) WYHKll;HJI F(P) HMeeT nOJIIOCbI nepBoro nOpJl)J,Ka: PI = 1 H P2. = -1-
Tor)J,a no TeopeMe 0 pa3JIO)KeHHH OPHrHHaJIOM )J,JIJI F(P) CJI)')KHT<PYHKll;Hn
f(t) = res F(P) . eP~ + res |
F(p)· ept . BbI'IHCJIHMBbI'IeTbI: |
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506
CJIe)J,OBaTeJIbHO, f(t) = ~ (et - e-t ) = sh t.
6) q,YHKll;IUI
rr03TOMY 3Ha'IeHIUIPI = 1 H P2 = -1 HBJIHlOTCH rrOJIlOCaMH BToporo rropH)J,Ka
cPYHKll;HH F(P). HaxO)J,HM Bbl'IeTbIcPYHKll;HH F(P) . ept : |
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HMeeT rrOJIlOCbI rrepBoro rrOPMKa PI = 0, P2 = 2i, P3 = -2i. Tor)J,a |
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res F(P)· ept = lim ~ = 1, |
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F (p) +;- f (t) = ~ - |
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= 1_ 1 cos 2t = 1(1 _ cos 2t) = sin2 |
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C nO.Mo'Ufb'lO meope.M'b/, 0 pa3.!UXHceHUU Ha11.mu opueUHa//,'b/, c//,eay'lO'W,ux U306paJICeHu11.:
8.2.61. |
F(P) = |
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8.2.63. |
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8.2.64. |
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F(P)= |
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507
AononHMTenbHble 3aAilYM
Hatimu OpUeUHa.fl,'bt c.fI,eayWW,UX u306pa:»eeHuti;
8.2.67. Q+~+ |
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8.2.71. |
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8.2.76.f(t) = t, g(t) = t.
8.2.78. f(t) = et , g(t) = et .
-r? +3p-3
8.2.68.
(p - 1)2(P - 2)"
_p4 + p3 _ 3p2 _ 1
8.2.70.
(p2 + 1)2p2
2e-3p
8.2.72.
p4 -1
8.2.74.- p - (e-:- P + 2e- 2p ). p2 + 1
8.2.77.f(t) = 3, g(t) = et .
8.2.79. f(t) = sin 3t, g(t) = sin 2t.
8.2.80.f(t) = t2 , g(t) = tet .
8.2.81.HafiTH CBepTKY <PYHKII,Hfi t· X(t) H (t - l)X(t - 1).
8.2.82.HafiTH cBepTKY <PYHKU:Hfi
sin(t - i)x(t - i) H sin(t-i)x(t-i)·
KOHTponbHble BonpOCbl M 60nee CnO)l(Hbie 3aAilHMR
8.2.83. |
.II;oKa3aTb, qTO eCJIH Ha OTpe3Ke [0, to] OpHrHHaJIbI f(t) H g(t) paBHbI |
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HYJIIO, TO CBepTKa f(t) *g(t) paBHa HYJIIO Ha OTpe3Ke [0,2to]. |
8.2.84. |
BepHo JIH, qTO eCJIH f(t) *g(t) == 0, TO: |
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a) o,n:Ha H3 <PYHKU:Hfi TO)K,n:eCTBeHHO paBHa HYJIIO; |
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6) B K3)K,n:ofi TOqKe o,n:Ha H3 <PYHKU:Hfi 06pamaeTcH B HYJIb? |
8.2.85. |
.II;oKa3aTb CBoficTBO KOMMYTaTHBHocTH CBepTKH: |
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8.2.86. MO)KHO JIH no CBepTKe 0,n:H03HaqHO BOCCTaHOBHTb CBepTbIBaeMble
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<PYHKU:HH? |
= sin t, HafiTH <PYHKU:HIO f(t). |
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8.2.87. |
3HM, qTO 1 * f(t) |
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HafiTH OpHrHHaJI H306p3)KeHHH |
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8.2.89. |
HafiTH OpHrHHaJI H306p3)KeHHH |
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508
§ 3. npM110>KEH~UI onEPAu.MOHHOrO MCl.fMCl1EHMSI
PeweHMe 3aAa'"lMKOWM A11R nMHeMHbix AMclJclJepeH ....ManbHblx ypaBHeHMM
MeTo,n;bI onepaD;HoHHoro HC'IHCJIeHHHy,n;06HO npHMeHHTb npH pemeHHH HeKO-
TOPbIX ,n;H«P<PepeHD;H8JIbHbIX ypaBHeHHit.
IIycTb 3a,n;aHO ,n;H<p<pepeHD;H8JIbHOe ypaBHeHHe (HanpHMep, 2-ro nopH,n;Ka) C nOCTOHHHbIMH K03<P<PHD;HeHTaMH: aox"(t) + alx'(t) + a2x(t) = f(t), r,n;e ao, aI,
a2 = const. Tpe6yeTcH HaitTH pemeHHe 3TOro ypaBHeHHH, y,n;OBJIeTBOpHIOmee HaqaJIbHbIM YCJIOBHHM: x(O) = Xl, x'(O) = X2. MCnOJIb3YH onepaD;HOHHoe HC'IHCJIe-
HHe, 3TO pemeHHe HaxO,n;HT CJIe,n;yIOmHM 06Pa30M. IIpe,n;noJIolKHM, 'ITOnpaBaH '1acTb ,n;aHHoro ypaBHeHHH HBJIHeTCH OpHrHHaJIOM. Tor,n;a H pemeHHe x(t) 3Toro YPaBHeHHH, y,n;OBJIeTBOpHIOmee 3a,n;aHHbIM Ha'laJIbHbIMYCJIOBHHM, TaKlKe 6y,n;eT opHrHHaJIOM. IIycTb x(t) -# X(P). MCnOJIb3YH TeopeMY 0 ,n;H<P<l>epeHD;HPOBaHHH OpHrHHaJIa, HaxO,n;HM H306palKeHHH npOH3Bo,n;HbIX, BXO,n;HmHX B ypaBHeHHe. B HameM CJIY'Iae:
X' (t) -# pX(P) - x(O) H |
X" (t) -# p2 X(P) - px(O) - X' (0). ,I1;aJIee, HaxO,n;HM H30- |
6palKeHHe <PYHKD;HH f(t) |
-# F(P). HaKOHeD;, npHMeHHH npeo6pa30BaHHe JIaILIIaca |
K o6eHM '1acTHMypaBHeHHH H nOJIb3YHCb CBOitCTBOM JIHHeitHocTH npe06pa3oBaHHH JIaILIIaca, nOJIY'IaeMonepamop'Hoe ypaS'He'Hue:
ao (p2 X(P) - px(O) - x'(O)) + al(pX(p) - x(O)) + a2X(p) = F(P).
OnepaTopHoe ypaBHeHHe (,n;pyroe Ha3BaHHe - u306plJ:)ICa1OU4ee ypas'He'Hue) HBJIHeTCH JIHHeitHbIM ypaBHeHHeM OTHOCHTeJIbHO HeH3BecTHoit <PYHKD;HH X(P). PemaH ero, HaxO,n;HM X(P) H, HaKOHeD;, no X(P) BOCCTaHaBJIHBaeM OpHrHHaJI x(t).
MHor,n;a MeTO,!I;bI onepaD;HOHHoro HC'IHCJIeHHHn03BOJIHIOT HaitTH pemeHHe JIHHeitHoro ypaBHeHHH C nepeMeHHbIMH K03<P<PHD;HeHTaMH. 9TO B03MOlKHO, HanpHMep, B CJIY'Iae,eCJIH <PYHKD;HH ao (t), aI (t), a2 (t) - MHoro'IJIeHbI,06bI'IHOHe CTapme nepBOit CTeneHH. O,n;HaKO HaxolK,n;eHHe pemeHHH B 3TOM CJIY'IaeCJIOlKHee, nOCKOJIbKY 6bIBaeT 3aTPy,n;HHTeJIbHO BOCCTaHaBJIHBaTb OpHrHHaJI no Hait,n;eHHOMY H306palKeHHIO (CM. npHMep 3.42).
PeweHMe MHTerpanbHblX ypaBHeHMM
~H'HmeepaAb'HtJt.MU ypaS'He'Htl.HMu Ha3bIBaIOTCH TaKHe ypaBHeHHH, B KOTOPbIX He-
H3BeCTHaH <PYHKD;HH x(t) CTOHT no,n; 3HaKOM HHTerpaJIa. |
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B HeKOTOpbIX CJIY'IaHXTaKHe ypaBHeHHH TOlKe MOryT 6bITb pemeHbI Cpe,n;CTBaMH onepaD;HOHHoro HC'IHCJIeHHH.K TaKHM ypaBHeHHHM OTHOCHTCH, HanpHMep, ypaBHeHHH BOJIbTeppa nepBoro H BTOPOro po,n;a, T. e. COOTBeTCTBeHHO ypaBHeHHH
!t k(t - r)x(r)dr = f(t) |
H x(t) = f(t) +!t k(t - r)x(r) dr. |
o |
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ilYCTb, HanpHMep, ,n;aHO HHTerpaJIbHOe ypaBHeHHe BOJIbTeppa BToporo po,n;a:
x(t) = f(t) +!t k(t - r)x(r) dr. o
509