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O o e ca oe e pay ecy ppoy epecy, yoy, ceoy ..

B eop ypae aoee pacpocpaee oyo aeaecoe pecaee cao. Bce aeaecx pecae cao ec a p ocoe py:

1) epepoe pecaee xoo ca opeee o oe pee (pc. 2.4, );

a) x(t) y(t) Oe ) ) ) y y y 0 0 t t t Pc. 2.4 B aeaecx pecae cao:

a o-cxea cce; epepoe; cpeo-epepoe;

cpeoe 2) cpeo-epepoe pecaee xoo ca ec aoa o pee epepo eec oo o ypo (pc. 2.4, );

3) cpeoe pecaee xoo ca aoa a o pee, a o ypo (pc. 2.4, ).

B peyae aoa caa o pee p cpeo-epepo cpeo pecaex oe poo oep opa, a a ocac ae caa oo cpee oe pee. Oao aoap ooy coc peax cce x p opeeex ycox coxpaec oa opa o cae, ec oce ece cpee oe pee. o coco eco a meopea omeuoa: ca, ocae ye c opae cepo, ooc opeeec co ae, oca epe epa pee t = Fc, e FC pa cepa caa.

Cc eope oeoa coco o, o, ec peyec epeaa ca, ocae ye f(t) c opae cepo, o ocaoo epeaa oee oee ae, ocae epe oe poeyo pee t = Fc. o ae epep ca oe ooc occaoe a xoe cce.

Maeaece pecae cao a pae ae ceo peayc e oy. o oyue oa eee ooo apaepo aoo-o ecoo poecca o aoy pecaeoo cooe. Ta, cceax c epec caa o oye oa eee ooo apaepo cooacooo epecoo caa o aoy epeaaeoo oacooo cooe. B cyae oy apoecoo caa paa a ocox a oy: aya oy yoa oy, oopa opaeec a acoy aoy. Ha pae ae ceo cpeac ceae oy ayo-aoa ayo-acoa, p o o o oy ec oe, pyo apa.

2.8 CHA. X B Haoee aco eop aoaecoo ypae coyc ceye ca.

1 E cao (pc. 2.5):

0 p t < 0;

x(t) = 1(t) = (2.16) 1 p t 0.

1(t) aaec ae ye Xecaa. Cpoo oop, y Xecaa ec epeayea, oao, ec, pepy, a cceyeo oee peo op e, peyae eo pacxo oaaeoo eeca ec cao c F1 o F2, o oop, o a xoe oea peaoa caoopa ca eo F2 F1, ec oce paoc paa ee, o a xoe peayec e cao.

Cepaa xapaepca eoo caa:

-i F(i) = e.

2 Ea yca y ea-y (pc. 2.6) o y, yoeopa cey yco:

0 p t 0;

1) (t) = pt = 0;

(2.17) 2) (t)dt = 1.

x t Pc. 2.5 E cao x t Pc. 2.6 E yc ea-y aa ae ye paa, oa oocc accy cypx y. y ec ae epeayey y oo peca a yc ecoeo ao eoc ecoeo oo ay, .e. a pee, oopoy cpec poyo yc c ocoae t oa, pao ee (pc. 2.7, a), ec t 0 a, o oa yca coxpaac pao ee.

Tae -y oo peca a pee eoopo y (pc. 2.7, ):

(t) = lim (t,) = lim. (2.18) (2t +1) x a) x ) = = = 0 t t Pc. 2.7 pecaee ea-y:

a poyo yc; (, t)-y oco coca ea-y oo oec ceye paeca:

0+ (2.19) (t)dt =1;

0-y ec eo ye:

(t) = (t); (2.20) x(t)(t)dt = x(0), (2.21) .e. epepo y oo pea oy opay.

oceee coooee, coy paccopee ye coca -y, oaaec cey opao:

0- 0+ 0+ x(t)(t)dt = x(t)(t)dt + x(t)(t)dt + x(t)(t)dt = x(0) = x(0).

(t)dt - - 0- 0+ 0Cepaa xapaepca ea-y: F(i) = 1.

Mey ye Xecaa ye paa cyecye c, paaea coooee:

(2.22) (t)dt = 1(), (t) = 1[t].

Ha pae caec, o a xo oea oaa -y, ec pe ec poyooo yca aoo ee pee epexooo poecca.

3 apoec ca (pc. 2.8, a) x(t) = A sint (2.23) coyec p cceoa cce aoaecoo peypoa aco eoa.

x a) x ) T b b c c a a t t d d x ) t t Pc. 2.8 apoec ca:

a o ca; pecaee apoecoo caa paee eopa; apoec ca co co a Cycoa apoec ca oo peca a paee eopa o A opy aaa oopa (pc. 2.8, ) c eoopo yoo copoc, pa/c.

a) ) x x 0 0 t t Pc. 2.9 Cye eeape y apoec ca xapaepyec a apaepa, a aya A; epo T; aa .

Mey epoo yoo copoc cpae coooe 2 = T =. (2.24) T Ec oea aac e y, o o xapaepyc ao oea (pc.

2.8, ), oopa o peeo oac xapaepyec opeo t, o oo ay paa paaax (pc. 2.8, ). epeo ocyecec o opye 2t =. (2.25) T Ha pae oye apoecoo caa coyec eepaop cycoax oea.

4 Cye eeape y.

y oocc y Xecaa paa c aaae, .e. 1(t ) (t ) (pc. 2.9), 0, t ;

(t - ) = pe, t =.

Bce coca -y coxpac, o acac e:

A + (t - )dt = 1;

(t - ) = ( - t) = (-(t - ));

x(t) (t - ) dt = x().

5 Ca pooo op x(t) (pc. 2.10, a).

a) ) ~ ) x x x ti ti 0 0 t t t Pc. 2.10 Ca pooo op:

a xoo epep ca; yc x(i);

cyepo yco, opeex ca x(t) o ca pooo op oo peca c oo -y. C o e eec poo oe pee t, cpoc co coo x(t) (pc.

2.10, ), cooecy ae caa oe pee t = ti ocoae ti.

~ o yc oo pa epe pey ea-y (t ti):

oa paa 1;

~, (t - ti ) = pa paa ti ;

coa paa ti, ~ .e..

xi (t) = x(ti )ti(t - ti ) ae y x(t) aopo yco (pc. 2.10, ), oo aca:

n ~(t) = )ti~ - ti ) x (t.

x(ti i=~ Ec eep n, ti d, (t - ti ) (t - ), o t+ x(t) = x()(t - )d. (2.26) Ca pooo op oo peca epe ee y, eo paee (2.26) ceye poeppoa o ac, coy coooee (t - ) = 1 (t - ), peyae eo o-ya ceyee coooee t + x(t) = x(0) 1[t] + x () 1(t - )d. (2.27) 2.9 TPEHPOBOHE AAH 1 B cceax aoaecoo ypae aac pae oec ca. ypoe aaa cea opex cce oyc papaoao ae x oec cao.

A ao ca aaec peyp i x ( t ) B ae cyecy pecae cao C ae ca oocc oco a peypx cao 2 xapaepc cepo cao coyec peopaoae ype. Cep epoecx cao xapaepyec opeee coca. eepoecoo caa oc oe cepao ooc.

A aoe peopaoae aaec peopaoae ype B a xapaep coca oaae cep epoecoo caa C o aoe cepaa xapaepca eepoeco y 3 B eop aoaecoo ypae coyc a aaee caape ca, oop oocc e cao, ea yca y eay, apoec ca.

A aa y aaec ea-ye B a a cceyeo oee oa ca e eoo caa C a apaepa xapaepyec apoec ca 2.10 TECT 1 Ca aaec peyp, ec eo aeaec pecaee ec:

A apaee aaa y pee.

B apaee aaa y aco.

C apaee aaa y pee aco.

2 Ca aaec epoec, ec o pecae coo:

A y pee yoeope yco f(t) = f(t + T), - t.

B y pee yoeope yco f(t) = f(t + T), t1 t t2.

C y aco yoeope yco f() = f( + W), -.

3 aoe peopaoa aaec peopaoae ype A F i = f (t)eitdt.

B F = f (t)e-itdt.

C F i = f (t)e-itdt.

4 Cepao ooc eepoecoo caa aaec ea 1 dA A F i =.

d 1 d B F i =.

dA dA C F i =, d e A ecoeo ae ay eepoeco y.

5 ye Xecaa aaec y:

0 p t < 0;

A x(t) = 1 p t 0.

B x(t) = 1 p t.

0 p 0 > t > t2;

C x(t) = 1 p 0 t t1.

6 ea-ye aaec y, yoeopa yco:

p t = 0;

A (t) = 0 p t 0.

0 p t 0;

B (t) = (t)dt = 1.

p t = 0; 0 p t 0;

C (t) = (t)dt = 0.

p t = 0; 7 aa y oocc cy eeap y A x(t).

B x(t ).

C x(t) + x().

8 Ca pooo op oo peca a:

t A x(t) = x() h(t - ) d.

t B x(t) = x() (t - ) d.

t C x(t) = x() h(t - ) () d.

9 Ca aaec apoec, ec A x(t) = Ah(t)sin t.

B x(t) = A(t)sin t.

C x(t) = Asin t.

10 Mey ye Xecaa ye paa cyecye c, paaea coooee A 1[t] = '(t).

B (t) = 1'[t].

C 1[t]dt = (t).

3 MATEMATECOE OCAHE ABTOMATECX CCTEM 3.1 OCHOBHE COCO MATEMATECOO OCAH. PABHEH BEH Maeaecoe ocae aoaeco cce ypae o ocae poecco, poeax ccee a e aea.

ocpoee oe cce ypae aaec c ye oea ypae cocae eo aeaecoo oca. B aece oea oe cya aapa, exooec poecc, pooco, pepe opac. Pae aeaecx oee oeo oycaaec x aaee. oe oca pae pe pao oea cce ypae oy oye o cocoo:

cepea, aaec, opoa cepeaoaaec.

p cepeao cocoe ypae oee oya ye ocao ceax cepeo (eo aoo cepea) ye caceco opao peyao eo pecpa epeex oea ycox eo opao cyaa (eo accoo cepea).

p aaeco oca ypae oee oya a ocoa oxecx aooepoce poeax poecco.

p cepeao-aaeco oxoe ypae oee oya aaec ye c ocey yoee apaepo x ypae cepea eoa.

p papaoe aeaecoo oca aoaecx cce ceye ya ocoe eoooece ooe eop aoaecoo ypae. o pee ceo cce oxo pee aa ypae, paccapa oeee oea peyopa poecce peypoa epapo aoc; oooc pee eoo eop aoaecoo ypae ccea cao paoopao eco ppo cece acpapoa aeaecx oee o opex ecx cce. poe oo, ccea paccapaec a e aoecyx ec opaoo eeo oaae cocooc epeaa ece oec opaoe ca oo, cpoo opeeeo apae;

a e ee cce paccapaec a peopaoae xooo oec xoy pea. Maeaecoe ocae a oex eeo, a cce eo cocaec, a pao, c po oye ypoe, yaoc oopx ac o y a cceoae cce ao oac, eo y oaeo oe cepeao poepe.

B oe cyae ypaeu ameamueco oeu oema uu cucme ynpaeu, ycmaauaue auoc ey xou u xou nepeeu, aamc ypaeuu ueu.

pae, ocae oeee cce peypoa ycaoec pee p ocox oecx, aac ypaeuu cmamuu.

pae, ocae oeee cce peypoa p eycaoec pee p poox xox oecx, aac ypaeuu uauu.

Bce oe peypoa oo pae a a acca: oe c cocpeooe oopaa, aa oopx ocaec ooe epea ypae, oe c pacpeee oopaa, aa oopx ocaec epea ypae acx poox. B aee paccapac oo oe c cocpeooe oopaa.

B aece pepa oo paccope oe c cocpeooe oopaa, ocae epea ypaee opoo opa (pc. 1.2) F(y, y', y", x, x') + f = 0, (3.1) e y xoa epeea; x, f xoe epeee; y', x' - epe pooe o pee; y" - opa pooa o pee.

p ocox xox oecx x = x0; f = f0 c eee pee xoa ea pae ocooe aee y = y0 ypaee (3.1) peopayec y:

F( y0, 0, 0, x0, 0) + f0 = 0. (3.2) oeoe ypaee (3.2) ec ypaee ca.

Caec pe oo xapaepoa c oo caecx xapaepc.

Cmamueco xapamepucmuo oea (cce) aaec acoc xoo e o xoo caeco pee.

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