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2 Hapaee e a paeop oea cpea. ee opaae o o aoo paeop pocxo o acoo cpee opy aaa oopa.

3 Boax y1 = 0, y2 = 0, .e. ocox oax, pocxo ocaoa e.

4 B cceax opoo opa aoe paeop epecea oc accc o p yo, a a p y2(t) = 0, =, a y1(t) = y(t) ocae coeo acya.

5 B epxx apaax oopao ococ opaaa oa ec dy ( t ) cea cea apao, a x - cpaa aeo, a a p y ( t ) = > 0 epeea dt dy ( t ) y1(t) = y(t) opacae, a p y ( t ) = < 0 epeea y1 (t) = y(t) yae.

dt 6 B o oe aoo ococ, e epeea y2(t) y f2(y1, y2) e pa y, aoa paeop ee oo oo opeeeoe apaee, cooecyee pooo ao oe, oya ceye, o aoe paeop e epeceac.

Haae yco epexooo poecca opee oopa aao o Ma aoo paeop.

Cooyoc aox paeop, cooecyx ce oo ao ccee aa yco, aaec ao nopmpemo cucme.

6.3.2 aoe oppe ex cce opoo opa oye ypae, ocax ao oppe cce opoo opa, eoxoo ccee epeax ypae (6.8) opoe ypaee oe a epoe c paccope pe t, peyae eo oya:

.

Peee oo ypae ae ceeco epax px a aoo ococ, o oop cpoc aoe paeop cce.

aoe oppe ex cce opoo opa accpyc o a ocox oe.

ea ccea opoo opa ocaec epea ypaee a (6.9) e y(t) - xoa oopaa cce; a0, a1, a2 - ocoe oe.

dy1(t) Ooa y(t) = y1(t), a = y2(t), oa dt, ypaee (6.9) oo aca e cce epeax ypae:

(6.10) Pae opoe ypaee a epoe, oya (6.11) peee oopoo ye ypaee aox paeop y2 = f(y1, c1, c2), (6.12) e ci - ocoe eppoa.

Boo ec pax cyae aox paeop acoc o ope xapaepcecoo ypae a2 s2 + a1 s + a0 = 0.

Cya op - e p a1 = 0, a0 > 0, a2 > 0: s1,2 = +i;

=. Ccea axoc a pae ycooc.

paeue cucme: a2y1"(t) + a0y1(t) = 0, eo peeue ueem u y1(t) = Asin(t + ), (6.13) oya y2(t) = y1'(t) = A cos(t + ). (6.14) pa y1(t) oaa a pc. 6.7.

Pc. 6.7 ao oppe a ep:

a - ococ ope xapaepcecoo ypae;

- epexo poecc; - ao oppe oye ypae aoo paeop pae (6.13) (6.14) oo apa caa, peyae oya ypaee:

. (6.15) Bpaee (6.15) pecae coo ypaee ca c oyoc A A. aaa pae A, oya ceeco aox paeop, oope e e epeceac e o ep aae oopa (pc. 6.7, ).

Hapaee e opaae o M ao ooe aoo ococ opeeec o ay y2. p ooeo ee y1 oe oo yeac, a p opaeo y2 - yeac, ceoaeo, ee opaae o a aoo ococ pocxo o acoo cpee, ooy eayxa epoec oea ccee cooecye a aoo ococ ayma aoa mpaemopu.

Ocoa oa cce ec eoepec epo aox paeop oc aae ep, a caa ccea aaec ocepamuo (.e. ccea e paccea ep, e pe).

Cya 2 op - oece e opaee eecee ac p a12 < 4a0a2; a1 > 0, a2 > 0, a0 > 0:

S1,2 = - i (pc. 6.8, a), = -a1/2a2, = (1/2a2) - ccea ycoa.

Peee ypae (6.9) ee :

y1(t) = Ae-t sin(t + ). (6.16) Pc. 6.8 ao oppe a yco oyc:

a - pacooee ope xapaepcecoo ypae;

- epexo poecc; - ao oppe Oya y2 (t) = y'(t) = Ae-t cos(t + + ), (6.17) e ;.

pae (6.16) (6.17) a aoo ococ apaepecoe ypaee cpae (c apaepo t). C a oopoo, cooecy ooy epoy oea, opaaa oa paec aay oopa, a a ae y1 ya epo oea caoc ee, .e. epexo poecc ee xapaep ayxax oea.

Ocoa oa aaec ycmou oyco.

Cya 3 op - oece e ooee eecee ac p a21 < 4a0a1; a0 > 0, a1 < 0, a2 > 0: s1,2 = + i.

o cya cooecye pacxoc oea ccee, .e. ccea ec eycoo. Peee ypae (6.9):

y1(t) = Aet sin(t + ). (6.18) Oya y2(t) = y'(t) = Aet cos(t + + ). (6.17) aoa oa, ac o aoo paeop, eopaeo yaec o aaa oopa.

Coco eycooo paoec cce cooecye ocoa oa, oopa aaec eycmou oyc (pc.6.9).

Ec peyae co yoo aoo oye ccea e coco paoec, o oa ye eopaeo yac o Pc. 6.9 ao oppe a eyco oyc:

a - pacooee ope xapaepcecoo ypae;

- epexo poecc; - ao oppe - eo o cpa aoo paeop, .e. ccee oae oeae poecc c opacae ayo.

Cya 4 op eecee opaee p a21 > 4a0a2, a1 > 0, a2 > 0, a0 > 0:

s1,2 = ;.

o cya cooecye aepoecoy poeccy ccee, caa ccea ycoa.

Peee ypae (6.9) y1(t) =. (6.20) Oya y2(t) =. (6.21) pae oac c epexo poecca a 1 2 cya pe c ypae y2 = s2 y1 y2 = s1 y1, oope oyac (6.20), (6.21) p s1 = 0 s2 = 0 (opaee ooo ope y).

Bce aoe paeop ac aao oopa - ocoy oy, aaey ycmou yo (pc. 6.10). Bpe e coco paoec eopeec pao ecoeoc.

Pc. 6.10 ao oppe a yco ye:

a - pacooee ope xapaepcecoo ypae;

- epexo poecc; - ao oppe Cya 5 op - eecee ooee p a12 > 4a0a2, a1 < 0, a2 > 0, a0 > 0:

s1,2 = .

B ccee ye aepoec poecc, oa eycoa. Peee ypae (6.9):

y1(t) =. (6.22) Pc. 6.11 ao oppe a eyco ye:

a pacooee ope xapaepcecoo ypae;

- epexo poecc; - ao oppe Oya:

y2(t) = y'(t) =. (6.23) aoe paeop apae o aaa oopa ecoeoc, .e. ec ccee eec ooee o coco paoec (aao oopa), o c eee pee oo ye eopaeo opaca.

Ocoa oa oc aae eycmou ye (pc. 6.11). o aao co cyae p epexooo poecca a 1 cooecy aoe paeop a 1, e pae paeop opeec ypae y2 = s1y1 y2 = s2y1. p epexooo poecca cooecy aoe paeop a 2.

Cya 6 op -- eecee e pae a p a1 > 0, a2 > 0, a0 < 0:

s1 = -1, s2 =. B o cyae ye eycoa ccea (p a0 = 0 - paa ycooc).

epexo poecc ccee ee aepoec xapaep, o ao oppe ee coepeo pyo .

ac ec cya, oa a1 = 0, , ya, o a0 < 0, ypaee (6.9) aec e ; (6.24) eppoae oo ypae ae:

2 y y 1 - = 1. (6.25) c ( c w ) Pc. 6.12 ao oppe a ceo:

a pacooee ope xapaepcecoo ypae;

epexo poecc; ao oppe Bpaee (6.25) pecae coo ypaee ceeca paocopox epo, oeceoe a oc. Acoa epo: y2 = y1.

aa aco coco pex aox paeop, .e. ocoa oa paccapaec a oa aox paeop.

Ocoa oa oc aae ceo, a aco a aoo ococ aac ceapapca cea (pc. 6.12).

o y ceapapca opaaa oa paec coco paoec, a o y py yaec o eo.

ac o o aoo paeop, opaaa oa o cee ocaoo ooo pee yaec o coco paoec a co yoo ooe paccoe.

Ceo ec eyco cocoe paoec, ae oa aae yco oo cooecy oe a ceapapce, aeee oyee po oy, o opaaa oa, oa a coce paeop, ye eopaeo yac o e o coco paoec.

6.4 OHTE CTOBOCT BEH Teop ycooc e a coaa aae aeo ea e pycc aeao Aecapo Mxaoe yo (1857 1918) c c aaa eeco exa.

a ccea, y oa eao (ec a ee e ecy ae oye) peao, ocaec epea ypae, peee oopx opeee paeop ee e.

ee aaec eoye, ec oo oyeo peyae paccope eapoao cce.

ee c yeo oye, oax peao ccee, aaec oye.

Heoyeoe ee aaec ycmou, ec ocaoo ae oye co yoo ao oo oyeoe ee o eoyeoo. Ec e oyeoe ee aeo ooec o eoyeoo p co yoo cax oyex, o oo aaec eycmou.

B eop ycooc cyecy pae o (ep), a o: opaa ycooc (ycooc o paeop), ycooc o yoy, acoeca ycooc ..

pee e epe opeee x o, eoxoo yo, o oaec o a oye. e oye oo pae a a a.

1 yce oye.

Pc. 6.13 ece ycoo oye:

a ycoe oyee; ee aoo pocpace Boyee aaec yc, ec oo ecye eee opooo poeya pee ( t) (pc. 6.13, a). yc ca oe, ec a pe t oopaa e yceae aeo ec. Bo cyae eo e aaec oeo ce opaae o M0 cce aaoo ooe M0 eoopoe pyoe ooee M0'. Tpaeop eoyeoo e cxo o M, a oyeoo M0' oaec o epo (pc. 6.13, ). Be yca caaec a ce e cce, xo o ecoa oo p pee t.

Ooa epe yi0 oopa o M0, i = 1, , n; epe y'i0 M'0,. p ao ce paoc oopa aa o acoo ee, .e. yoeope yco, e eoopoe ocaoo aoe ooeoe co.

Pc. 6.14 Hepepo ecye oye Ma oyeue aaec aoe ycoe oyee, oopoe ae a c aaoo ooe opaae o cce.

Ma oye cooecy ae, e ee, e ee oye.

2 Hepepo ecye oye.

Tae oye ecy a ccey e oo aa oe pee, o oceye (pc. 6.14). Ha ep aec, o ye ax oye ceae oee o o, a a o e oee oy opy, e yce. Ho a pae oaaec e a. Cce, ycoe p ycx oyex, yco p epepx; eycoe p epo e - eyco p opo.

po oo ec o a, o epepoe oyee oo peca e oceoaeoc yco, .e. papea ec pa x(t) a yc eoc dt, ooy aee paccapac yce oye.

6.5 OCHOBHE B CTOBOCT 6.5.1 Opaa ycooc Boc oe -opecoc eoyeoo e. C o e paccapaec paeop eoyeoo e M0M cpoc poe p payco, oc oopoo ec a paeop.

Caec, o paeop oyeoo e ao ooec o paeop eoyeoo e, ec oa eo e -opecoc eoyeoo e ( - ao). Boyeoe ee cxo o M '0 (pc. 6.15).

cooc - o coco e, eee aece, a e oece xapaep. ooy p opypoe o ycooc aa paa oooc oopa co aoe, o pa oyeoo e e a o-pecoc eoyeoo e p o ae. Ec aa oooc cyecye, o ee ycoo, ec oa ocycye, o eycoo.

Pc. 6.15 o "Opaa ycooc" oop, ccea oaae opao ycooc, ec p o oo oopa aoe ooe o y aee pae, o paeop oyeoo e e a -opecoc eoyeoo e, o oceee aaec yco. Ec e oopa aoe e, o eoyeoe ee eycoo.

oe opao ycooc ee cyece, pa eocao, opaa pee eo peoc. p opao ycooc oyeoe ee oe aeo oac o eoyeoo.

Ec ae paeop , o o M M ' yc c pa copoc, o c eee pee paccoe ey oe oaac o (pc. 6.16), .e.c yi oopa o M, a y'i - M ', o p a opao ycooc oe oaac, o e (yi - y'i) cay o. B c c oc oe ycooc o yoy.

Pc. 6.16 o ycooc o yoy 6.5.2 cooc o yoy ee aaec yco o yoy, ec oo > 0 oo yaa co = () > 0 aoe, o epaeca p t = t0 ceye epaeco cex t > t0.

Cc o ycooc o yoy coco o, o ee ycoo, ec p ocaoo ao aao ce M'0 o M0 oa M' oceye e ocaoo a M (pc. 6.16). Ec e oopa aoe () e, o ee eycoo.

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