Introduction to Laplace Transforms October 25, 2017

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1 Iroduco o Lplc Trform Ocobr 5, 7 Iroduco o Lplc Trform Lrr ro Mchcl Egrg 5 Smr Egrg l Ocobr 5, 7 Oul Rvw l cl Wh Lplc rform fo of Lplc rform Gg rform b gro Fdg rform d vr rform from bl d horm pplco o dffrl quo Exmpl for homogou d ohomogou quo Rvw L l d x d x x l Equo dx x dx x Soluo for o-gr = J x + J x Soluo for gr = = J x + Y x fd l fuco bl d from Excl gr ol or MTL boh gr d o-gr J x l Fuco J x l Fuco of h Fr Kd for Igr Ordr J = J = for >= = = = x Yx l Fuco Y x l Fuco of h Scod Kd of Igr Ordr Y pproch mu f x pproch for ll x = = = 5 Lplc Trform fo Trform from fuco of m, f, o fuco complx pc, F, whr complx vrbl Th rform of fuco, wr F = L[ f] whr L do h Lplc rform Lplc rform dfd h followg grl: L[ f ] f d F Hv bl of F f 6 ME 5 Smr Egrg l Pg

2 Iroduco o Lplc Trform Ocobr 5, 7 Smpl Lplc Trform f F f F!/ + x x+/ x+ / co / + co / + ddol rform h / pp 6-67/8-5 of Krzg h / h do coh / 7 Fdg Lplc Trform lhough w c u h rform dfo o fd rform, w uull rl o bl Smpl gro: fd L[ ] L[ ] = Such gro for xmpl ol; w c fd uch mpl rform h rform bl 8 Wh Lplc Trform? Smplfd chqu for olvg l vlu problm IVP U rform bl o g rform lbld F for uul fuco, f Trform dffrl quo wh dpd vrbl o lgbrc quo wh lgbrc vrbl d l codo Tbl ud o rform quo rm from f o F d vc vr Wh Lplc Trform co d Trformd OE pc mpuld o g ol F form vlbl rform bl U rform bl o g rform from rm F bck o f rm You ow hv h oluo! Smpl xmpl: fd for d/d + k = wh = Smpl Exmpl Fd for d/d + k = From rform bl Ld/d = Y d L[k] = ky Trformd quo of d/d + k = Y + ky = Fd Y b lgbr: Y = / k From rform bl for Y = /, = -, o oluo = -k c Trform Propr Lr Thorm L[f + bf ] = L[f ] + bl[f ] Fr hfg horm If L[f] = F h L[ f] = F rvv rform for L[f] = F L[df/d] = F f L[d f/d ] = F f f L[d f/d ] = F - f - f f - f - ME 5 Smr Egrg l Pg

3 Iroduco o Lplc Trform Ocobr 5, 7 Lr Thorm Sm of Thorm L[f + bf ] = L[f ] + bl[f ] Proof of lr horm L[f + bf ] = f bf d f d b f d = L[f ] + bl[f ] Imporc: Th Lplc rform of OE wh co coffc h um of h rform of ll rm h OE Fr Shfg Thorm If L[f] = F h L[ f] = F G proof of fr hfg horm b modfg grl for F o g F F = F = = = L[ f] E.g. L[ω] = F = ω / ω h L[ ω] = F = ω /[ ω ] rvv Trformo rvo for f = df/d Lf = Lf rvv rform of dffr ordr L[df/d] = F f L[d f/d ] = F f f L[d f/d ] = F - f - f f - f - 5 Trform Noo U lowr c lr for dpd vrbl orgl dffrl quo Tm,, h uul dpd vrbl If h OE u f, u F for h oo of h rformd vrbl Smlrl u Y for h rformd vrbl f h orgl vrbl Trform bl uull u f d F 6 Solvg ffrl Equo Trform ll rm h dffrl quo o g lgbrc quo For dffrl quo w g h rform Y = L[] Smlr oo for ohr rformd fuco h quo R = L[r] Solv h lgbrc quo for Y Ob h vr rform for Y from bl o g Mpulo of rqurd o g from Y quo o rform bl 7 ffrl Equo Exmpl Trform ll rm h dffrl quo o g lgbrc quo For + + = w g h followg: Y + [Y ] + Y = Solv h lgbrc quo for Y + + Y = Y = + + ' Y 8 ME 5 Smr Egrg l Pg

4 Iroduco o Lplc Trform Ocobr 5, 7 G Ivr Trform From rform bl Krzg co 6., h do, p., r d for b b F f b b b F f b b W hv ' Y b ppl Ivr Trform ' Y ' ' Soluo vld for l codo bou Ivr Trformo U rform bl M b bl o u h fr hfg horm dcud rlr Mhod of prl frco wll b dcud lr h lcur U cod hfg horm Scod hfg horm u dfo of Hvd u fuco d rc dl fuco dcud x lcur Nohomogou OE Exmpl Look ohomogou c for m OE lzd prvoul + + = r Trform ll rm h quo d rrrg o olv for Y Y + [Y ] + Y = R = L[r] + + Y = R + + Y = + R + + Nohomogou Exmpl II Solv + + Y = + R + + for Y R ' Y Q Y R Q ' Q Q, H or W lo ud for Q clld h m or rfr fuco Nohomogou Exmpl III Scod rm m homogou c R ' Y ' ' Krzg, Tbl 6., No. & ME 5 Smr Egrg l Pg

5 Iroduco o Lplc Trform Ocobr 5, 7 ME 5 Smr Egrg l Pg 5 5 Nohomogou Exmpl IV Pck om r d g corrpodg R o g xmpl oluo Pck r = o R = / + ' ' R Y o fd /++b + rform bl 6 Prl Frco Mhod o covr frco wh vrl fcor domor o um of dvdul fcor domor Exmpl /++b Wr /++b = /+ + /+b Mulpl b ++b d qu coffc of lk powr of = + b = for rm d = b + for rm d + = for rm 7 Prl Frco II + = for rm d = b + for rm Solvg for d gv = - = /b Rul: /++b = /[b + ] /[b + b] Th cull mch bl r Follow m bc proc for mor complx frco Spcl rul for rpd fcor d complx fcor 8 Prl Frco Rul Rpd frco for rpd fcor omplx fcor Pur mgr fcor complx fcor wh = Exmpl: ppl Prl Frco Spl xpro o foud bl o compo complx fcor, + Mulpl b orgl domor oug Exmpl Equ coffc of lk powr = + + rm = rm = rm = + + rm Solv for,,, d

6 Iroduco o Lplc Trform Ocobr 5, 7 ME 5 Smr Egrg l Pg 6 Solv for,,, d Gu lmo for ugmd mrx ckolv for,,, d Wr Y G Trform ' Y Wr Y G Trform II ' Y ' co 5 Lplc Trform Soluo Uul oluo r wh umrcl vlu of d l codo Th umrcl vlu mplf h rul Soluo prd hr vld for d l codo ' co 6 Ohr pplco W c ppl h o m of quo for Trform ll quo from o Y Solv mulou lgbrc quo for ch Y G vr rform for Wll codr m of quo h x cl prod

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