"105 Jag = Tnverse LAPLACE TRANSFoR Mm SE _EARIACE _TRANSror wv “Le = FCA) then, p(t) the inverne (i) & Ws dendled by LL rca) . Linearity Pruspody : for) + bo] = al[reay]4 be fee] Opllace Trompos lin % 1 Taventt_Loplloce Thanipotorn S AsOme Aardond. puncson : g EG)=! DE (Bjgs)= Aina 5 Gale.) = einai a U(ck)= ot qe (sap) = Ainhab 5 ECL) = sinha a (ai): ¢ 3 Ge) = Bo (noua gq e( cosat Ly il wee) * re Ge “ a J (Be) = coshat “rind the inweue (oploceTangorm op, thu péllowing punctitin ) A c (2) ou at{ A] sal)=a J] HATE GAP 10K | = Hoy [pS ‘ chase See TE w+? + ott y 4° = 5 7 “RH TRO pte SEV + Sten g +505 “4 = “ds“| PA-> + HA-1g9], = a) ok 4 ee aa | nee : = pels | wey 1 = ue in aie) Fa fy = WE ae] + RET Wy ARR ~ aa ATH 3 A+ 8. p - 5 Ain (5 +18 Ainhat 3 I" ing m I> & 1 og 4 a TC SS i Pai) 2 UY xy = tet + Ainet = Los(syt 4b rin(2}t ~ Latha 1 + 6Ainh3t : 5) ey era APH16 Sosket + Aintph + sodas! ag) tla = afin Vv = LL Anlt 4 | Bate 4+ JA-50 AF TE ops - evo eS = (em | clang wy) - w [ae] *: #4 50L of] % oth(2 jt ee z POCA ~ 9 cothut + 6 Aint = A wothays - spiminly oe oo ae hSe ee ; 4 lat 7 ae + 38 Baal) basal ao et - 2c + 3? S ee a 2 |, gent Laplace Tondyonmn by COrmyplating the howe the paupesdy hak , Lepow! = F(A) ELem™ pty] = “phis impls woe F(A-0) eqs muthod ; es that ELF CAI] = pet) —-O ad e'[ra-a)] = & Ht) —@ © fe'fr(s-a] = o& cee vu " e'Lrtasaa] = Find the inverse lop J Ate Ry AHN comidan dinemindlon ; by complating A aces reathoo, AHA HIE EGP -HA EHH IB = (ARP 49 . rate G2 +4 Add & Aub 2 In Rumen 3 mY w&-ad+ee4+2 } G-2E49 ule F(A 7 su pots pce fay gee tea (4-29 Hew as2 and 4-274 ate if Ath = ee l a | See oy eo Ad Az ee. (ae + Hainst)MH (22h Sa ~ J Ate ARATE AS) AMG EAHE a (Pa ene)-14e = (Caentet te Age ek aml = yf GEeD-1+e 3) ! @-3y * [as] How a=3 and £34 t y Bt I eog 1 oe fo ed 4 2 eh i = ot tH { 2h go A i GND) ~ Be 4 Find the invoug 4 loplnce [radi totter sehen sre ie te tng bs Gut Sinai) az! and At 24 ete i arr at yf A ac ore Kai AH ee. ( cost + Aint) nv " n “J Atl AeBK+I e, At 3 SF | Gee a] Farr i case 4 2 [Goo] oy cfGeies 3 Cc @-n? = oe +2] at 4a we fee (14et) = ef (14et) +8 y AR ka 1 = A(At2)(A4S) + B(ALDAva IW EC Aa nave) eS pub A=-1 t= ACa42)(-43)+0+0 => [an Yopub 5°35 te (arn Gave) es Vp lamas - eT] ef) +a bs) 2 Ain F bet net, ye P mano - ger Laat p= ACRE (BAI) KH) Wa ge la y ont = Gers) 5 G+) Rel ice sft je 4h! 4) af eT 4] 2 (actin * 2 [a] 2 [Ail + 3 Axl = pets +4 art 2 2 23 FA Ae 2 eee ee - (4-2) (A439 A-2 At3 (2) (4-2) qA= Acaes)™ pa A583 a(pys Ales a = Ws | pat 82-35 apg) 2c (872) exxply 8h 475 1 Sol -# e| 4] 3 : [a te }+8 os F Gy ap [ee 2s Lr essa : ei oh oe e+ Bete 4) 25 es 5S Az e met ett petty 25 25 is > Trvene Thariponm op Lagat om and inode renders cee Giver F(A), we nied +8 pind c ‘[pat] = 1® fe? yell: have. tht ae ey on “F) we pronedg tL. pe Faye | Cmjog (aa a yen FOAY 109(4 ) fy. eval = tiple) (a sbog (4~ -a)- ~ Log (4 b) 7 asp) pnt 4 § mutiny pith G1) on b-S . . \ fe! AA oa ae es) co te _ -F A) ay Tare i oo wa(erta = ML da a] pee) = ee ptt) = @ -& a ftuy= t"(%) Rapp nb Ag xW by Gi) Rove oe way oO. 4 on bs, A) = fa FMA) = oe k FAY = an 4A> ard 2 FMA) aw a ofr] = cf fo. ty) = pinot pin = inate *E7 eS SES 2 RR — Ayo? ay (485) shan? F(A = toy (448 ) 2 HAY = bag (A740?) ~ Lag( 4b?) App wort 4 “p= PAL PA FMA ts aR EAS Bs Fay ae = yo? =f 8 Bs [-Fva] = ey Hy - £2] wp) = Ratbt = 2usab ao > twee Joplace Thaniporm a aed Bb Lote l= F(A) zp ULptey] = Fla) then L[ Spade] x 2 PLEO] = (y0.at eT EO] = Si cf ecy] a =, FA) A> Trarapoum ep th ee ! I Zl af 724 oo ie at “| 2] = f ae ott “(aia]= fel ae! = fee a clark 4 +f % fin3t ok | aes 2 aa), ae ia, fe (ete ot) - 0+ ae H _ aa a 5 et betoverg pends a 4 [ese]? zo = cl [us 3t - cso) q = aa -1] =, Lawoist q3) t A(AMH) + oof 1.) ok Tae) ae ce J - s wept ot + = -caret | = pe = = 1 fot ia i Th : Us F(4) Unit dtep Function in Invest ; xp eters FO) then EL (a) ult-ad ds, & L = efetra] iz pitrou lea) snd ha Dot eplos Worgorn ah the polewing. parton: 4 ele em (Aa fer] borer gray] = 5) vf 4 |= trae = po) ! Rae h wehoue, fe ear] = plt-a)ult-a) fe"s(Ara)} = p(t )uce-rr) = o9s[3-w)] ule-m) = osf- (arr-3t)] ut) = w0h(3rr-3t) (4-11) = -M3btult-m) |!ENCrUv]= 5) +2 os ce! Hy (Aa, soe [Aa] + [Fae] = COLRE + Ainat = f tL) whey iofcok pray Te p(t-a)ult-a) clea ye p(t-aT)u(t-er) Aerh h (corece-am) + ina (t-21)} ult-en) [-cos(urr-at) —Ain(urr-ay)) ult-2r) [cot(urr-2t) —Ain(urrat)] ult) [unet +Ainet] ult-eir) " 3 [Se] Feeley 9 FL] 10 How a=! and At 74 ~ bel 7 “ls a] 2 € £ = ple) efe* = Ree". - wlt-3) 2 } = [Fe] way" X > Convolution : 8 tuition, ¥ s Th HA ond aft) ave woo partion’ d eh kash i _ 4 and g(t) u Brora 'y page on para gt) = fy m wg tt aude wo > Cenvetudion Theorem + ay e'[roo] = pee) ond E c'fren-6(4)) = SL pngeen fearl=g then, oe rBllurns 2 oxprosed a yO sas ae = Wenki J The given pundtion i (4) and GCA) weidepme ELECANI= Ct) and EMfo 1=g%) al we only the easton Taser era of the form L'fr(a)- 64 = Ly (w) gta) de 3 Evoluode the conveldtion integral +o olstrin the spat {Ne - oad Heng “the orutbibion, Thastene 7 ange % mee pobloing i he 4 ' AAP +08) td FU)e L ord GlA= a Ae c'[Ftay] = ely] c'foray] = ct vee| = r= p(t) = Anak = 9 tt)r clos [5 ft) get-w) di (a0?) [stn] * £ 1. nate) dae ssn ass | et web F(AY= and GCA) = wa Arta? eletal= E'fotgge] = “ath = 6) witocnd = © yee] * senor = gft) any Larwdlution thesar, i OL EL ey] «J Spe Meee ae we Ainh fon = L[ us8(A-8) -wilarsy] a nf fort os) — | cos (at) | du uso a “Ut : ko ty Dok | Aan (Qos ~ ok) cat | RO ° so a Ainok jutat? — (a -9)| Ro? Ro sta ant | 2a Ainak — cia3) _4 Maayan) het F(AY = and 6(4) = a A743? rf e*Crra)] = [tel s oat =) i int = g (4) etecat= eft ]s «“t g 4 “ [eacesn = { coygu. #in(eu) du Med esta. 4inB = 1 [Ain(Ae) ~rin(-8) } Butlt-uy Sutt—u ps = }f [in (aust) ~ vin(uu-t) } ebu Cok 2) sea i : 2 Zz = 1 mae : Zone = 4 [ ~oisty osat _ [ott + cout 2 2 2 He z | sa a ‘ 2 — Eon e +f eRh + = a = A] ust — iat LL att cosse) | ue Thanporms of the Desavativer . we dosive on oniphention we Kyi] and hence eluece Hh oxphenion poh Hyer] Aas fy], purthan , we wie the prerciple Mathumnatical Trduction+- ettobtigh "the povult por Hye. ct De,= cools rlyeel = ie rat y(t) Sues ufeffuda | oe kt ah ~ Fytey. ot ¢- : sgorky wo]. fs ota) dt 2 fo-tyeo)] vafe ot yte) th -ylo) FALL wl [yt] = i{tyiaal'4 eS ALL ye) ] - yo) fA ALlyay] + yloyh yo) Ga: ately ly] ~Ayle).- y'0)|—-® 7 ily") = Axlyle)] ~ Ayo) - 49 (0) -y"Co) ~®@ Tn gensral, ! a Lyn] = AP LLy lt] - 4" aes Ayila)--~ 9“ No) > Aplition 84 LOE wang Loplace Thanijonm m_[ Initial ving Rue: dep: The iver dijpourtial cyan i onpvusscd in the ntdlien yi» yt) yh) jor the dasivaives. Sipe: We take fapilace Thana{onm On both Jide of the given Alps: We ue the exphenion for ily] , yo] and 49 6% Ake + dubatiteb the given initial conditiom & imply +0 Obtain Lye] fe pundion %p A" Zep S + we Wind the invowe 16 plitain y Ct - Rrdblema + dato yrs yy'a ay = 0 unde the conditions gl)=% Yo) <, : by ey) tahoe THON Om given OF us eepeuned. in yh Ge) ey) + By(t)=0 : : TTaktng Lapdace Tranyyoten on bs HLyney] + wily co] 4 aily)] = ilo) ArLy(t)] - AYylo -yilo)+ wf arlyct] ~ycoly +3LLylt)] =o | Aubytitule y@) rs) tondation $e above egiun A aking the initial Landakions, ist obtain : ALlylty] LK) —) ¥ yaely ced ~ (0) + 3LLyl] = 0 yer { At thé 434 =P = Ef 1? * lead By completing Aquose method , ALEK E BLS (tbs Hole +3 = Gr2F-5 m8 aa | TE is in the fotm * 2 leave] = lea] 1 yl) = ee ey a" sromfege Y= See UNa the cOndlitions J fe ung foplace THangyovm w(t) 4 ayltt) ay) = 6te* yeh aulyay] + yc) = elltet] gly] — Ayo) ~yleo) 4 pfArlyce ~yoy+ Hyco] ylol=o=y'loy - = 6hlet.t] ' “Les inthe porn Lem cay |= er cbt - pry) + 2aUyco] + a = Gels) RNS ztyter {Aaeaery = ar eile t] 5 (=+ 3 Ly 7ebey | oi | ll rab = by . GHY (A24244 1) -ipaui) = i: (5 ] Be ont mot : Att = [ seerten | ee (44)? 7 “| =| La HOw a=) AN PS = 6e* ef) é ae r = 6%. 43 3) = etyif 2 a ae 7 way + hy = with, yo) =Os i ys wy) wy Ct) = ot My] ay KLy Cel] A atl ww) = Ke) Ue 4 fy] q wf aL laced] ~ylod} th ily) = ‘ie J ~ Ayle) -y!Co) kegel. ALi) + ysely ced) +ulyor= Cle) by se ay o¥® Liye ys ey ¢ a tee yh) = a ! Genin cen Pearl le Gry? (AH) -L GH) (at2)?-4 \ ; Sr = A a 2B c Galant ~ 2 ae “Gene P= AlAt2)? 4 BC AW (442) ¢ clan) Raeta c2ey see cwoepy 4? I aC ; @=1te!. | ors hose]7 = _« EF _ : ae ~ itt 3 | gamer o elas! _ | ft] S fate] o:e* a ott _ ite et -8* [144]