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(s +1)3(s - 2) aoe opaee pacaaec a pocee po:

s2 + 2 A1 A2 A3 B = + + +.

s +1 s (s +1)3(s - 2) (s +1)2 (s +1)3 - paa ac oceeo pae poc oey aeae, yco paeca cee oya:

s2 + 2 = A1(s +1)2(s - 2) + A2(s +1)(s - 2) + A3(s - 2) + B(s +1)3.

paeca oeo p cooecyx ceex s eo pao acx acaec ccea aepaecx ypae:

A1 + B = 0;

A + 3B = 1;

A3 - A2 - 3A1 + 3B = 0;

- 2A3 - 2A2 - 2A1 + B = 2, peee oopo ae A1 = 2/9; A2 = 1/3; A3 = 1; B = 2/9. Ta opao, s2 + 2 2 1 1 = - + - +.

9(s +1) 9(s (s +1)3(s - 2) 3(s +1)2 (s +1)3 - 2) pe opaoe peopaoae, acaec paee opaa:

s2 + 2 2 1 1 L-1 = - e-t + te-t - t e-t + e2t.

(s +1)3(s - 2) 9 3 2 3.9 EPEATOHA H Oo ocox xapaepc oea ypae, coyeo eop aoaecoo ypae, ec epeaoa y, acaea epax peopaoa aaca.

epeamoo yue oea aaec ooee peopaoaoo o aacy xoa oea y(s) peopaoaoy o aacy xoy x(s) p yex aax ycox.

epeaoa y opeeec oo ype coca cce, ec ye oecoo epeeoo ooaaec:

y(s) W (s) =. (3.36) x(s) epeaoa y xapaepye ay oea oo o opeeeoy aay, caey ope xo oea ope xo (pc. 3.13).

Ec oe ee ecoo xoo xoo, o o xapaepyec eco epeao y, opee oope oo eocpeceo, oyc opeeee (3.36).

) a) x1 W1(s) y y x W(s) xW2(s) ) W11(s) yxx2 yW12(s) W22(s) Pc. 3.13 pep pax oeo:

a c o xoo o xoo; y xoa o xoo; y xoa y xoa pep 3.4 yc a xo oea oaec ca x(t) = 1(t), a a xoe caec ca, ocae ye y(t) = 2 e2t.

1 opeee epeaoo y eoxoo opee x(s) = ; y(s) = s s + 2s oa epeaoa y W (s) =.

s + a epeaoe ypaee, epeaoa y ooc xapaepye ay eoo oea. Ec aao epeaoe ypaee oea, o oye epeaoo y eoxoo peopaoa epeaoe ypaee o y(s) aacy oyeoo aepaecoo ypae a ooee.

x(s) B oe cyae epeaoe ypaee oea pecaec e an y(n)(t) + an-1y(n-1)(t) +...+ a1y (t) + a0 y(t) = = bmx(m)(t) + bm-1x(m-1)(t) +...+ b1x (t) + b0x(t), (3.36, a) e an, , a0; bm, , b0 ocoe oe.

oce peopaoa o aacy p yex aax ycox oya:

ansn y(s) + an-1sn-1y(s) +... + a1sy(s) + a0 y(s) = = bmsmx(s) + bm-1sm-1x(s) +... + b1sx(s) + b0x(s), (ansn + an-1sn-1 +... + a1s + a0)y(s) = (bmsm + bm-1sm-1 +... + b1s + b0)x(s), oa y(s) bmsm + bm-1sm-1 +... + b1s + bW (s) = =. (3.37) x(s) ansn + an-1sn-1 +... + a1s + aEc eca epeaoa y oea, o opaee xoa oea y(s) pao poee epeaoo y a opaee xoa x(s):

y(s) = W(s) x(s). (3.38) oce ac ec e o oe, a oa opa ac pee epeaoo ypae oepaopo ope.

Ta opao, epeaoa y paa ooe yx ooo:

B s W s =, A s e B(s) = bmsm + bm-1sm-1 +... + b1s + b0 ; A(s) = ansn + an-1sn-1 +......+ a1s + a0 y.

peax ecx oeo oo oe a xapaepy ocoeoc o a, o cee ooa B(s) cea ee paa cee ooa A(s), .e. m n, a o lim W (s) = 0.

s epeaoa y ae ao ooao caa c pee xapaepca.

Ec eec paee epexoo y, ceoaeo, xoo ca x(t) = 1(t) x(s) =, xoo ca y(t) = h(t) y(s) = h(s), oa epeaoa y s paa h(s) W (s) = = sh(s). (3.39) x(s) (3.39) oe oyeo paee epexoo y epe peopaoae aaca:

W (s) h(s) =. (3.40) s Ec eco paee ecoo y, o xoo ca x(t) = (t) x(s) = 1, xoo ca w(t) , ceoaeo, w(s) W (s) = = w(s), (3.41) x(s) .e. paee epeaoo y ec e o oe, a peopaoae aaca o ecoo y.

pep 3.5 yc oe ocaec epea ypaee y (t) + 3y (t) + 4y(t) = 2x(t); y(0) = y (0) = 0. Ha h(s) w(s).

pe peopaoae aaca: s2 y(s) + 3sy(s) + 4y(s) = 2x(s), opeee epeaoy 2 2 y W (s) =. epexoa y h(s) = ; h(t) = L-1 2.

s2 + 3s + 4 s(s2 + 3s + 4) s(s + 3s + 4) 2 Becoa y w(s) = ; w(t) = L-1.

s2 + 3s + 4 s2 + 3s + 3.10 TPEHPOBOHE AAH 1 Maeaeca oe oea ypae cce ypae ycaaae aoc ey xo xo epee. Paa ypae ca ypae a. caoeo, o pae o eco ppoe oe ypae oaa eoop o epa ocac oo ypae c o pe aea.

A ae ypae aac ypae ca o pecae coo caeca xapaepca B ae ypae aac ypae a C a ypae ocac oe ypae: paec peepyap, epeca eoc, epep oepec xec peaop ooo epeea 2 O acco cce, oope paccapae eop aoaecoo ypae o ee caoape cce, oec py cyepo. Ocoo aae ye aecoo oee x cce ec yee pacca xoo ca oo ecoo xooo caa, .e. pacca ay cce. C o e coyc aece xapaepc. Oco pee xapaepca, oope, a pao, oya cepeao, c epexoa y ecoa y.

A a oaa, o ccea ec eo cceo B ae xapaepc oocc aec xapaepca C o pecae coo cxea pacea a c oo peex xapaepc 3 Oco aeaec aapao, coye eop aoaecoo ypae, ec peopaoae aaca, c oo oopoo acaec ocoa aeca xapaepca oea ypae epeaoa y.

A ae opeeee peopaoa aaca. Copypye ocoe coca.

B ae epax peopaoa aaca epeaoe ypaee 4y (t) + 2y (t) + y (t) + 2y(t) = sin t, y(0) = 0, y (0) = 1, y (0) = 0.

C aa xapaepca aaec epeaoo ye 3.11 TECT 1 aoe ypae ec ypaee a A F y, y, 0, x, x + f = 0.

B F y, y, y, x, x + f = 0.

C F y0, 0, 0, x0, 0 + f = 0.

2 a epea ypaee ocaec aa ax oeo ypae, a epeca eoc, xec peaop ooo epeea dy(t) A T + y(t) = k x(t).

dt d y(t) dy(t) B T12 + T2 + y(t) = k x(t).

dt dtdy(t) dx(t) C T + y(t) = k.

dt dt 3 Maeaeca ac pa cyepo coco ceyx coooe y (t) yi xi (t) ;

xi A i i y x(t) y x(t).

y (t) = yi xi (t) ;

xi B i i y x(t) y x(t).

y (t) = yi xi (t) ;

xi C i i y x(t) = y x(t).

4 aa aeca xapaepca aaec epexoo ye A Pea cce a e cyea ca.

B Pea cce a -y.

C Pea cce a apoec ca.

5 a coooee ycaaaec c ey epexoo ye ecoo ye A h(t) = w (t).

t B h(t) = w(t)dt.

C h(t) = w(t) + w (t).

6 ay c ycaaae epa ae A Mey xo xo cao pooo op.

B Mey epexoo ye ecoo ye.

C Mey xo cao pooo op xo cao.

7 aoe peopaoae aaec peopaoae aaca A x(s) = (t)e-stdt.

x -st B x(s) = x(t)e dt.

-it C x(s) = x(t)e dt.

8 aa xapaepca aaec epeaoo ye A Ooee peopaoaoo o aacy xooo caa peopaoaoy o aacy xooy cay.

B Ooee xooo caa xooy p yex aax ycox.

C Ooee peopaoaoo o aacy xooo caa peopaoaoy o aacy xooy cay p yex aax ycox.

9 Ec eca epeaoa y, o epexoa y opeeec a W (s) A h(l) = L-1.

s B h(l) = L-1 sW (s).

C h(l) = L-1 W '(s).

10 ao epa aaec epao ae A y(t) = - ) x(t)d.

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

w(t C y(t) = x()d.

w(t) 4 ACTOTH METO CCEOBAH HEHX CCTEM 4.1 EMEHT TEOP H OMECHOO EPEMEHHOO oec co aaec co, opeeeoe coooee z = a + i b, e a b cooeceo ecea a ac ca. Taa opa ac oecoo ca aaec aepaeco. Ha oeco ococ, oopaax Re (ecea ac) Im (a ac), oecoe co eoepec pecaec eopo (pc. 4.1); oo oe opaeo ae opx oopaax M (oy) (aa) acao oaaeo ope: z = Mei, e M a eopa, coeeo aao oopa c oo z; - yo ey ooeo e eceo oc eopo z, pe ooe apaee caec apaee ocea po acoo cpe.

Im z b M a Re Pc. 4.1 opaee oecoo ca Tpe opa ac oecoo ca pooepeca, a a ei = cos isin, z = M cos iM sin.

Bce cocae oecoo ca ca ey coo cey coooe (pc. 4.15):

b M = a2 + b2 ; = arctg ; a = M cos; b = M sin.

a p ce a (apyea) ca eoxoo ya, ao apae axoc oa z. He poc opy, o oop cee a coc Im z opeee ocpoo ya, paoo arctg (pc. 4.2).

Re z b I apa: z1 = a + ib, 1 = arctg ;

a b b a II apa: z2 = -a + ib, 2 = arctg = - arctg = + arctg ;

- a a 2 b -b b 3 a III apa: = -a - ib, arctg = + = - arctg ;

z3 3 = arctg - a a 2 b -b b 3 a IV apa: z4 = a - ib, 4 = arctg = -arctg = + arctg.

a a 2 b Im z2 zb 1 a a Re b zzPc. 4.2 Opeeee a acoc o pacooe eopa z ypoe oepa a oec ca oeo a, o 1 = ei0; -1 = ei; i = ei / 2; - i = e-i / 2.

Ha oec ca poo e e apeece oepa (coee, ae, yoee, eee), o a ece. Coee ae oee yoo poo a oec ca, aca aepaeco ope:

z3 = z1 z2 = (a1 ib1) (a2 ib2) = (a1 a2 ) i(b2 b1), a yoee eee a ca, aca oaaeo ope:

z3 = z1z2 = M1ei1 M ei2 = M1M ei(1 +2 ) ;

2 z3 = z1 / z2 = M1ei1 / M ei2 = M1 / M ei(1 -2 ).

2 Ec apye y oecoe co, o y ec ye oecoo epeeoo. Hapep, y W(s), s = + i.

Ta opao, oo caa, o ye oecoo epeeoo aaec eoop oepaop (pao), coaco oopoy a) i Im ) W(s) s W(1) W(0) Re Pc. 4.3 opeee y oeco epeeo oe oo ococ oecoo epeeoo cac cooece oa pyo ococ oecoo epeeoo (pc. 4.3).

Ec y oocc accy aaecx y (epepa, aa, o cy epepyea), o aa y oec pa oopoo oopae, oco coca oopoo c ceye:

1 oo oeco ococ s oopaaec pyo oeco ococ W(s) (pc. 4.4).

2 ecoeo a yo oopaaec ao e ecoeo a yo, y p o coxpac (pc. 4.4).

3 ecoeo a peyo oopaaec ao e pa ey ecoeo a peyo. Hapaee oxoa yo coxpaec. Bype oac ooo peyoa peopayec o ype oac pyoo peyoa (pc. 4.4).

i Im ) b a) c a s = + i W(s) A C B Re Pc. 4.4 oopoe oopaee 4.2 ACTOTHE XAPATEPCT Bay po p oca ex cce pa acoe xapaepc, xapaepye pea oea (cce) a apoec ca.

Ocoo acoo xapaepco ec ayo-aoa xapaepca (AX), oopa oe opeeea epe oopoe oopaee.

i Im W(s) S 1 = 0 1 = Re W(i ) 2 > Pc. 4.5 opeee AX Ayo-aoo xapaepco aaec oopoe oopaee o oc ococ ope xapaepcecoo ypae a oecy ococ ayoaoo xapaepc (pc. 4.5), pe caa a oc oopaaec oopa AX, paa e oyococ ope xapaepcecoo ypae oopaaec o ype oac AX.

Ayo-aoa xapaepca ec oeco ye, ooy oa oe , a a oeca y, pecaea oaaeo ope W (i) = M ()ei() (4.1) aepaeco ope W (i) = Re() + i Im(). (4.2) Moy M() oaaeo ope ac AX aaec anumyo-acmomo xapamepucmuo (AX), a aa apye () aaec ao-acmomo xapamepucmuo (X).

ecea ac ayo-aoo xapaepc Re() aaec eecmeo acmomo xapamepucmuo (BX).

Ma ac ayo-aoo xapaepc Im() aaec uo acmomo xapamepucmuo (MX).

Mey ce aco xapaepca cyecye c (pc. 4.1). a o x, oo opee pye, .e.

M() = Re2() + Im2(), (4.3) Im() () = arctg, (4.4) Re() Re() = M () cos (), (4.5) Im() = M ()sin (). (4.6) 4.3 CB PEOPAOBAH AACA PE a eco, a ea caoapa ccea aoaecoo ypae ocaec ooe epea ypaee, oopoe oepaopo ope ee (ansn + an-1sn-1 +... + a1s + a0)y(s) = (bmsm +bm-1sm-1 +... +b1s +b0)x(s), (4.7) e y(s) = y(t)e-stdt - peopaoae aaca y y(t).

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