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Bxoo ca y(t), oyac eee aoo poecca, ec aoee oo xapaepco aecx coc cce, ooy opeeee oo caa, a ye oeaoc, ec ocoo aae eop peypoa. ec caoc ayao e ye aecx coc cce c oo peex xapaepc.

3.6 EPEXOHA BECOBA H 3.6.1 epexoa y oye epexoo y aece caapoo caa coyec ea y pee (2.16). Taoo poa oec cooecye, apep, cpoc ee apy cceax peypoa (oa oopa ccee peypoa).

h ) x(t) a) h() qx S t qx t Pc. 3.10 epexoa xapaepca xecoo peaopa:

a cyeaoe oece; pa paoa epexoo yue aaec aaecoe paee pee eoo epeaoo ypae (3.8) p xoo cae x(t) = 1(t) yex aax ycox, .e.

an y(n)(t) + an-1y(n-1)(t) +...+ a1y (t) + a0 y(t) = b01(t), (n-1) y(0) = 0; y (0),..., y (0) = 0. (3.12) puo paoa aaec pea oea (cce) a eoe cyeaoe oece p yex aax ycox.

Ha pae pa paoa opeeec cepea ye coyec aece cxox ax aaa cea cce aoaecoo ypae cceyeoo oea.

ec ceye ec o po opao aa. pa aaa (aaa o) aaec opeee pee epeaoo ypae c aa aa yco. B opao aae peyec occao oe epeaoo ypae o eco epao po, apep, epexoo y. Peee opao aa pecae aey cooc cece ee eoppeoc ec cyecye cea aeaec aapa. Ta, apep, ec peoo, o epexoa y ocaec peee ypae epoo opa a1y (t) + a0 y(t) = b0x(t), x(t) = 1(t), y(0) = 0, Ty (t) + y(t) = kx(t), b0 ae k = ; T =, o opeee oea k oe yce T ocoa a0 apee.

y() Bcae y'(t) = 0 , ceoaeo, y() = k x(), oya oe yce k =, x() a a x() = 1; y() = h(), o k = h().

opeee ocoo pee T cxooe ypaee eppyec peeax o 0 o :

T y (t)dt = - y(t)]dt = - h(t)]dt.

[kx(t) [h() 0 0 paa ac oceeo pae ec e o oe, a oa S o cepeao co po paoa (pc. 3.10, ), oa oo aca: T h() = S, S oya T =.

h() 3.6.2 Becoa y oye ecoo y, ee ae aa unyco nepexoo yue, aece caapoo caa coyec -y (2.17):

0 p t ;

(t - ) = (t)dt = 1.

p t = ; Ta opao, ecoo ye w(t) aaec pea cce a -y p yex aax ycox.

Ha pae ecoy y oex cyax oo oy cepea ye eca peo. Ca, o a xo oea oaa y, ec pe ec yca aoo ee pee epexooo poecca.

pepo oe cy cepe o c ecoo y xecoo peaopa (pc. 3.4), eoc oeo cceoa. B aece xooo caa peaop ao aec op paceo eeca (apep, ep). epe eoopoe pe o eeco oc a xoe, pe eo oepa epoaao opacae, a ae yae pacee eeco aec (pc. 3.11).

oaae a xo yc pecae coo pey ea-y, a a eo oa oa o e paa S. ooy oye ecoo y cepeao c epexo poecc oppy ye ee eo opa a ey oa xooo oec S.

x a) ) w S S t t t t Pc. 3.11 epexoa xapaepca xecoo peaopa:

a -y; ecoa y Mey pee xapaepca: epexoo ecoo y cyecye aoe ooaoe cooece, oopoe opeeec cey opao:

t w(t) = h (t); h(t) = w()d.

Becoy y oo oy a peee epeaoo ypae an y(n)(t) + an-1y(n-1) (t) +...+ a1y (t) + a0 y(t) = b(t);

y(t) = y (0) =... = y(n-1) (0) = 0.

p pee oox ypae ea-y epeo aae yco, b ec n = 2, o a2 y (t) + a1y (t) + a0 y(t) = 0; y(0) = 0; y (0) =.

a3.7 HTEPA AME epa ae coyec opeee xoa oea y(t) p pooo xoo cae x(t) ecx h(t) o w(t).

peoaaec, o a xo oea, ocaeoo ecoo ye w(t), oaec ca x(t) (pc. 3.12, a), opooe ocae oopoo ao . 2.8.

Ec pea oea a (t ti) ooa epe w(t ti) (ecoa y), a pea ~ ~ a (t - ti ) epe w(t - ti ) (pea ecoa y), o a ocoa pa ~(t) cyepo oo aca xoo ca a yc x :

~ ~ yi (t) = w(t - ti )ti x(ti ).

~ x a) y ) ti yi ti 0 t t Pc. 3.12 pecaee xooo (a) xooo cao () aea xooo caa x(t) aopo yco, coa oopx coaae c cooecy oopaa (pc. 3.12), ooe aca pea a cyeay ~(t) y x a ocoa pa cyepo n n ~(t) = y (t) = ~ y ~i w(t - ti )ti x(ti ).

i=0 i=~ ~ Ec eep ycpe ti 0, p o ti ; n ; (t - ti ) (t - ); w(t - ti ) w(t - ), a ti d, e epep apaep, oaa c aoo yca, o ooaeo oyae:

y(t) = - )x()d. (3.13) w(t oceee ypaee aaec epao ae (ypaee cep), opaa c ey xoo, xoo oea eo ecoo ye.

o cy ea ecoa y ec a oea, oopa oaae, a oo a co e a oe ycoe oyee, oaoe a eo xo oe pee = 0.

ecoo cca ecoo y epx pee eppoa oe aee a t, a a eooo peca peay ccey, oopo a xoy oopay aco oe pee oaa e oye, oope oc oceye oe pee.

Ec poec aey opye (3.13) t = =, d = d, o oo aca cepy opyy y(t) = - )w()d. (3.14) x(t Ec pecae xooo caa cooa e opyy (2.26), a (2.27), o epa ae acaec epe epexoy y:

t dx() y(t) = x(0)h(t) + - ) d, (3.15) h(t d t dx(t - ) y(t) = x(0)h(t) + h()d.

d 3.8 PEOPAOBAHE AACA Oco aeaec aapao, oop coyec eop aoaecoo ypae, ec cea eo paoo aaa, a aae oepao eo, ocoe oopoo e yoaoe peopaoae aaca.

3.8.1 Opeeee peopaoa aaca peopaoae aaca aaec peopaoae y x(t) epeeo t y x(s) pyo epeeo s p oo oepaopa, opeeeoo coooee L{x(t)} = x(s) = x(t)e- stdt, (3.16) e x(t) opa y; x(s) opaee o aacy y x(t); s oeca epeea s = + i.

opya (3.16) opeee poe peopaoae aaca. Booo a aaeoe opaoe peopaoae aaca, ooee o opae a opa. Oo opeeec coooee c+i L-1{x(s)} = x(t) = x(s)estds, (3.17) 2i c-i e c accca cxooc y x(s).

oca y, cpeaxc a pae, cocae a cooec ey opaa opae. opae eoopx aoee aco cpeaxc y eop ypae pee a. 3.1. Ec e y ocycye ae, o ee opaee oo oy eocpeceo, oyc coooee (3.16).

pep 3.1 Tpeyec a peopaoae aaca o y x(t) = eat.

Coaco opeee peopaoa aaca (3.16) ee 1 -at -(s+a)t x(s) = e-stdt = e e dt = - s + a e-(s+a) = s + a.

0 Ta opao, e-at.

s + a Taa 3.Taa peopaoa aaca Opa opae Opa opae e e 1 (t) 1 8 sint s2 + s 2 1 cost s s2 + 3 t e-t sint (s + )2 + s2 tn s + n! 4 (n = 1, 2, e-t cost (s + )2 + sn+1 ) (1- e-t ) 1 e-t s(s + ) s + 1 1(t - a) e-as t et (s + )3 s tn e-t (s + )n+pooe peee peopaoa aaca oycoeo e, o opaee eoopx y oaaec poe x opao p oepa, ax a eppoae, epepoae a opae poe, e cooecye oepa a opaa.

3.8.2 Coca peopaoa aaca p cooa peopaoa aaca eoxoo a pe eo coca, eoope x opypyc cey opao.

1 Teopea eoc: x ecex oe-cx ocox A B eo oa opao cooecye aa e oa opae (3.18) Ax1(t) + Bx2 (t) Ax1(s) + Bx2(s), e x1(t) x1(s); x2(t) x2(s).

2 Teopea oo: yoee apyea opaa a oe ocooe ooeoe co po ee apyea opae x(s) a o e co :

1 s x ( t ) x. (3.19) 3 Teopea ayxa: yoee opaa a y eat, e a oe eceoe oecoe co, ee a coo ''ceee" eaco epeeo s:

eat x(t) x(s - a). (3.20) 4 Teopea aaa: oo ocooo > x(t - ) e-sx(s). (3.21) 5 Teopea epepoa o apaepy: ec p o ae r opay x(t, r) cooecye opaee x(s, r), o f (t, r) f (s, r). (3.22) r r 6 Teopea epepoa opaa: ec x(t) x(s), o x (t) sx(s) - x(0), (3.23) .e. epepoae opaa coc yoe a s eo opae a x(0).

B acoc, ec x(0) = 0, o x'(t) s x(s). pe eopey eoxooe oeco pa, oya x(n) (t) snx(s) - sn-1x(0) - sn-2x (0) -...- x(n-1) (0). (3.24) Ec x(0) = x (0) =... = x(n-1) (0) = 0, o x(n) (t) sn x(s), (3.25) .e. p yex aax aex n-paoe epepoae opaa coc yoe a sn eo opae.

7 Teopea eppoa opaa: eppoae opaa peeax o 0 o t po ee opae a s:

t x(s) x(t)dt. (3.26) s 8 Teopea epepoa opae: epepoae opae coc yoe opaa a (-t) :

-tx(t) x (s). (3.27) 9 Teopea eppoa opae: eppoa opae peeax o s o cooecye eee opaa a t, .e. ec epa x(z)dz cxoc, o s x(t) x(s)ds. (3.28) t s 10 Teopea yoe opae: ec x(t) x(s), y(t) y(s), o cepe y t x y = y(t - ) d (3.29) x() cooecye poeee opae xy x(s) y(s). (3.30) 11 Teopea yoe opao: poee opao cooecye cepa opae +i y(t) x(t) = y(s)x(s) = x(z) y(s - z)dz, (3.31) 2i -i e = Re z.

12 Teopea o oeo aao aex y:

lim x(t) = lim sx(s) ; (3.32) t slim x(t) = lim sx(s). (3.33) t0 s 3.8.3 Peee epeax ypae O aex pee oepaooo cce peopaoa aaca ec peee ex epeax ypae c oco oea, oop a pa ocac paccapaee cce aoaecoo ypae.

Peee epeaoo ypae o cyae caaec ceyx ao:

1) peopaoae ypae o aacy;

2) ocae pee oac oecoo epeeoo s;

3) epexo oac eceoo epeeoo ye opaoo peopaoa aaca.

pep 3. a2 y (t) + a1y (t) + a0 y(t) = b01(t) ;

y(0) = y'(0) = 0.

peopaye aoe ypaee o aacy:

a2s2 y(s) + a1sy(s) + a0 y(s) = b0 1/ s, oya by(s) =.

s(a2s2 + a1s + a0 ) yc oo a2s2 + a1s + a0 = 0 ee op s1 s2, oa, a ye oaao e, oo aca C0 C1 Cy(s) = + +, s s - s1 s - se C0, C1, C2 eoope oe, opeeee eoo eopeeex oeo:

b0 b0 bC0 = ; C1 = ; C2 =.

s1s2 s1(s1 - s2 ) s2 (s2 - s1) oyc aa opaoo peopaoa aaca, axo y(t) = C0 + C1es1t + C2es2t.

oyeoe paee y(t) ec peee eoo ooeoo epeaoo ypae opoo opa p xoo cae x(t) = 1(t), .e. e , a epexoo ye eoo oea opoo opa.

3.8.4 Paee a pocee po a o pepa 3.2, peee epeaoo ypae, oyeoe c cooae peopaoa aaca, pecae coo paoay po. oee opaoo peopaoa oyey po eoxoo pao a pocee po, oyc cey pao.

po n-1(s) M (s) = (3.34) n(s) aaec pao paoao po, ec opo ce ee, e opo aeae. paoe po (3.34) eoxoo a op ypae n (s) = 0.

Ec ope ece, o ey cooecy po a A.

s - sEc op ecee paoc k, o cooecye cya poe A1 A2 s Ak sk -+ +... +.

s - s1 - s1)2 (s - s1)k (s Ec op oeco copee, o A1s + B.

(s2 + as + b) Ec op oeco copee paoc k, o A1s + B1 A2s + B2 Ak s + Bk + +... +.

(s2 + as + b) (s2 + as + b)2 (s2 + as + b)k Ta opao, po (3.34) oo peca e n-1(s) A1 A2 Ak = + +... + + n(s) (s - s1) - s1)2 (s - s1)k (s B1 B2 Bm + + +... + +... + (s - s2) - s2)2 (s - s2 )m (s (3.35) C s + Dp C1s + D1 C2s + Dp + + +... + + p (s2 + a1s + b1) (s2 + a1s + b1)2 (s2 + a1s + b1) Fqs + Eq F1s + E1 F2s + E+ + +... + +...

(s2 + a1s + b1) (s2 + a1s + b1)2 (s2 + a1s + b1)q oe A1,..., Ak; B1,..., Bm; C1,..., Cp; D1,..., Dp; F1,..., Fq; E1,..., Eq axoc eoo eopeeex oee. B o cyae paa ac (3.35) poc oey aeae oyaec paeco yx poe, y oopx aeae pa, ceoaeo, o pa ce. paeca ocex cocaec ccea aepaecx ypae opeee eecx oeo, oopa peaec ec eoa pee ex aepaecx cce.

p opeee opaa o oyeoy opae oyc cey opya cooec:

A Aes1t ;

s - sA k - A t es1t ;

(k -1)! (s - s1)k a - t As + B B - Aa / e sin t b - a2 / 4.

Acost b - a2 / 4 + s2 + as + b b - a2 / s2 + pep 3.3 Ha opa, ec opaee.

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