Th Lplc Trnform Dfiniion nd propri of Lplc Trnform, picwi coninuou funcion, h Lplc Trnform mhod of olving iniil vlu problm Th mhod of Lplc rnform i ym h rli on lgbr rhr hn clculu-bd mhod o olv linr diffrnil quion. Whil i migh m o b omwh cumbrom mhod im, i i vry powrful ool h nbl u o rdily dl wih linr diffrnil quion wih diconinuou forcing funcion. Dfiniion: L f b dfind for. Th Lplc rnform of f, dnod by or L{f }, i n ingrl rnform givn by h Lplc ingrl: L {f } f d. Providd h hi impropr ingrl xi, i.. h h ingrl i convrgn. Th Lplc rnform i n oprion h rnform funcion of i.., funcion of im domin, dfind on [,, o funcion of i.., of frquncy domin *. i h Lplc rnform, or imply rnform, of f. Toghr h wo funcion f nd r clld Lplc rnform pir. or funcion of coninuou on [,, h bov rnformion o h frquncy domin i on-o-on. Th i, diffrn coninuou funcion will hv diffrn rnform. * Th krnl of h Lplc rnform, in h ingrnd, i uni-l. Thrfor, h uni of i h rciprocl of h of. Hnc i vribl dnoing complx frquncy. 8 Zchry S Tng C- -
Exmpl: L f, hn, >. L{f } f d d Th ingrl i divrgn whnvr. Howvr, whn >, i convrg o. Exmpl: L f, hn, >. [Thi i lf o you n xrci.] Exmpl: L f, hn, >. L{f } d d Th ingrl i divrgn whnvr. Howvr, whn >, i convrg o. 8 Zchry S Tng C- -
Dfiniion: A funcion f i clld picwi coninuou if i only h finily mny or non whovr coninuou funcion i conidrd o b picwi coninuou! diconinuii on ny inrvl [, b], nd h boh on-idd limi xi pproch ch of ho diconinuiy from wihin h inrvl. Th l pr of h dfiniion mn h f could hv rmovbl nd/or jump diconinuii only; i cnno hv ny infiniy diconinuiy. Thorm: Suppo h. f i picwi coninuou on h inrvl A for ny A >.. f K whn M, for ny rl conn, nd om poiiv conn K nd M. Thi mn h f i of xponnil ordr, i.. i r of growh i no fr hn h of xponnil funcion. Thn h Lplc rnform, L{f }, xi for >. No: Th bov horm giv ufficin condiion for h xinc of Lplc rnform. I i no ncry condiion. A funcion do no nd o ify h wo condiion in ordr o hv Lplc rnform. Exmpl of uch funcion h nvrhl hv Lplc rnform r logrihmic funcion nd h uni impul funcion. 8 Zchry S Tng C- -
Som propri of h Lplc Trnform. L {}. L {f ± g} L {f } ± L {g}. L {c f } c L {f }, for ny conn c. Propri nd oghr mn h h Lplc rnform i linr.. [Th driviv of Lplc rnform] L {f } or, quivlnly L { f } Exmpl: L { d } L {} d In gnrl, h driviv of Lplc rnform ify L { n f } n or, quivlnly L { n f } n n Wrning: Th Lplc rnform, whil linr oprion, i no mulipliciv. Th i, in gnrl L {f g} L {f } L {g}. Exrci: U propry bov, nd h fc h L { }, o dduc h L { }. b Wh will L { } b? 8 Zchry S Tng C- -
Exrci C-.: U h ingrl rnformion dfiniion of h Lplc rnform o find h Lplc rnform of ch funcion blow... 6. co. in.* iα, whr i nd α r conn, i. 6 8 Ech funcion blow i dfind by dfini ingrl. Wihou ingring, find n xplici xprion for ch. [Hin: ch xprion i h Lplc rnform of crin funcion. U your knowldg of Lplc Trnformion, or wih h hlp of bl of common Lplc rnform o find h nwr.] 6. 7 d 7. d 8. in 6 d 9. L f b coninuou funcion nd b i Lplc rnform. Suppo f for. Show h i qul o h r of h rgion undr h curv y f,. b Vrify hi fc by compring h vlu of wih h r undr h curv obind h uul wy i.. by ingrion for ch of h funcion,, nd. 8 Zchry S Tng C- -
Anwr C-.:.. 6.. 9 α. i α α No: Sinc h Eulr formul y h iα co α i in α, hrfor, L{ iα } L{co α i in α}. Th i, h rl pr of i Lplc rnform corrpond o h of co α, h imginry pr corrpond o h of in α. Chck i for yourlf! 6. 7 7. 8. 6 9. S in h dfiniion f d o how h f d A. 8 Zchry S Tng C- - 6
Soluion of Iniil Vlu Problm W now hll m h nw Sym : how h Lplc rnform cn b ud o olv linr diffrnil quion lgbriclly. Thorm: [Lplc rnform of driviv] Suppo f i of xponnil ordr, nd h f i coninuou nd f i picwi coninuou on ny inrvl A. Thn L {f } L {f } f Applying h horm mulipl im yild: L {f } L {f } f f, L {f } L {f } f f f, : : L {f n } n L {f } n f n f f n n f n f. Thi i n xrmly uful pc of h Lplc rnform: h i chng diffrniion wih rpc o ino muliplicion by nd, n lil rlir, diffrniion wih rpc o ino muliplicion by, on h ohr hnd. Eqully impornly, i y h h Lplc rnform, whn pplid o diffrnil quion, would chng driviv ino lgbric xprion in rm of nd h rnform of h dpndn vribl ilf. Thu, i cn rnform diffrnil quion ino n lgbric quion. W r now rdy o how h Lplc rnform cn b ud o olv diffrniion quion. 8 Zchry S Tng C- - 7
Solving iniil vlu problm uing h mhod of Lplc rnform To olv linr diffrnil quion uing Lplc rnform, hr r only bic p:. Tk h Lplc rnform of boh id of n quion.. Simplify lgbriclly h rul o olv for L{y} Y in rm of.. ind h invr rnform of Y. Or, rhr, find funcion y who Lplc rnform mch h xprion of Y. Thi invr rnform, y, i h oluion of h givn diffrnil quion. Th nic hing i h h m -p procdur work whhr or no h diffrnil quion i homognou or nonhomognou. Th fir wo p in h procdur r rhr mchnicl. Th l p i h hr of h proc, nd i will k om prcic. L g rd. 8 Zchry S Tng C- - 8
8 Zchry S Tng C- - 9
Exmpl: y 6y y, y, y [Sp ] Trnform boh id L{y 6y y} L{} L{y} y y 6L{y} y L{y} [Sp ] Simplify o find Y L{y} L{y} 6 L{y} L{y} 6 L{y} 9 6 L{y} 9 L{y} 9 6 [Sp ] ind h invr rnform y U pril frcion o implify, L{y} 9 6 b 9 6 b 9 b b b 8 Zchry S Tng C- -
Equing h corrponding cofficin: b 9 b b Hnc, L{y} 9 6. Th l xprion corrpond o h Lplc rnform of. Thrfor, i mu b h y. Mny of h obrvn udn no doub hv noicd n inring pc ou of mny of h mhod of Lplc rnform: h i find h priculr oluion of n iniil vlu problm dircly, wihou olving for h gnrl oluion fir. Indd, i uully k mor ffor o find h gnrl oluion of n quion hn i k o find priculr oluion! Th Lplc Trnform mhod cn b ud o olv linr diffrnil quion of ny ordr, rhr hn ju cond ordr quion in h prviou xmpl. Th mhod will lo olv nonhomognou linr diffrnil quion dircly, uing h xc m hr bic p, wihou hving o prly olv for h complmnry nd priculr oluion. Th poin r illurd in h nx wo xmpl. 8 Zchry S Tng C- -
Exmpl: y y, y. [Sp ] Trnform boh id L{y y} L{ } L{y} y L{y} L{ } [Sp ] Simplify o find Y L{y} L{y} L{y} L{y} L{y} 8 L{y} [Sp ] ind h invr rnform y By pril frcion, 8 L{y} b c. 8 Zchry S Tng C- -
8 Zchry S Tng C- - 8 c b c b c c c c b b c b c b c b 8 b c c L{y} 8. Thi xprion corrpond o h Lplc rnform of. Thrfor, y. No: L { n }! n n
Exmpl: y y y, y, y [Sp ] Trnform boh id L{y} y y L{y} y L{y} L{ } [Sp ] Simplify o find Y L{y} L{y} L{y} L{y} / L{y} / L{y} L{y} 6 6 [Sp ] ind h invr rnform y By pril frcion, L{y} 6. Thrfor, y. 8 Zchry S Tng C- -
or h nx xmpl, w will nd h following Lplc rnform: L {co b} b b L {in b} b, >, > L { co b} b b L { in b} b, >, > No: Th vlu of nd b in h l wo xprion dnominor cn b drmind wihou uing h mhod of compling h qur. Any irrducibl qudric polynomil B C cn lwy b wrin in h rquird from of b by uing h qudric formul o find ncrily complx-vlud roo. Th vlu i h rl pr of, nd h vlu b i ju h bolu vlu of h imginry pr of. Th i, if λ ± µ i, hn λ nd b µ. 8 Zchry S Tng C- -
8 Zchry S Tng C- - 6 Exmpl: y y y co, y, y [Sp ] Trnform boh id L{y} y y L{y} y L{y} L{co} [Sp ] Simplify o find Y L{y} L{y} L{y} L{y} / L{y} / L{y} L{y} [Sp ] ind h invr rnform y By pril frcion, L{y} 6 which corrpond o [ ] in co in co y
8 Zchry S Tng C- - 7 Exmpl: ind h invr Lplc rnform of ch i 8 Rwri : 9 Anwr: in 9 co f ii U pril frcion o rwri : 6 Anwr: f
Appndix A Som Addiionl Propri of Lplc Trnform I. Suppo f i diconinuou, hn h Lplc rnform of i driviv bcom L {f } L {f } lim f. II. Suppo f i priodic funcion of priod T, h i, f T f, for ll in i domin, hn L {f } T T f d. Commn: Thi propry i convnin whn finding h Lplc rnform of diconinuou priodic funcion, who diconinuii ncrily hr r infinily mny, du o h priodic nur of h funcion would mk h uul pproch of ingring from o unwildy. III. L c > b conn, h im-cling propry of Lplc rnform h L {f c } c c. IV. W hv known h L { f }. Tking h invr rnform on boh id yild f L { }. Thrfor, f L { }. 8 Zchry S Tng C- - 8
Th i, if w know how o invr h funcion, hn w lo know how o find h invr of i ni-driviv. ormlly, L { f } u du. V. Similrly, dividing by corrponding n ingrl wih rpc o. L { f u du }. Commn: Thrfor, dividing funcion by i indpndn vribl h h ffc of ni-diffrniion wih rpc o h ohr indpndn vribl for Lplc rnform. Th wo propri IV nd V r h counr pr of h muliplicion-corrpond-o-diffrniion propri n rlir. VI. Afr lrning h fc h L {f g} L{f } L{g}, on migh hv wondrd whhr hr i n oprion of wo funcion f nd g who rul h Lplc rnform qul o h produc of h individul rnform of f nd g. Such oprion do xi: L { f g d } L{f }L{g}. Th ingrl on h lf i clld convoluion, uully dnod by f * g h rik i h convoluion opror, no muliplicion ign!. Hnc, L{f * g} L{f } L{g}. urhrmor, L{f * g} L{g * f }. 8 Zchry S Tng C- - 9
8 Zchry S Tng C- - Exmpl: L f nd g, find f * g nd g * f. f * g d d 6 6 6 g * f d d 6 6 6 In ddiion, obrv h, by king h invr Lplc rnform of boh id h propry VI, w hv d g f L {L{f }L{g}}. In ohr word, w cn obin h invr Lplc rnform of impl funcion of h i ilf produc of Lplc rnform of wo known funcion of by king h convoluion of h wo funcion of. Exmpl: ind h invr Lplc rnform of. ir, no h L{}L{in } Thrfor, i invr cn b found by h convoluion ingrl ling f nd g in :
8 Zchry S Tng C- - co co co in d co. Th nwr cn b ily vrifid uing h uul invr chniqu. Exmpl: ind h invr Lplc rnform of. Sinc i produc of h Lplc rnform of nd, i follow h f * d d.
Exrci C-.: ind h Lplc rnform of ch funcion blow.. f. f co 6 in 6. f. f 7 6. f in in b f in co 6. f co b f co 7. f co b, b f in b 8. f co 9. f co α β b f in α β. f in Hin: in α inα in α 9 ind h invr Lplc rnform of ch funcion blow... 6 8.. 7. 6. 8 6 6 7. 8. 8 9. α β 8 Zchry S Tng C- -
U h mhod of Lplc rnform o olv ch IVP.. y y, y. y y, y. y 6y in, y. y y, y 6, y. y y y, y, y. y y y, y, y 6. y 8y y, y, y 8 7. y 6y y, y, y 8. y y 8co 8, y, y 9. y y y 8 6, y, y. y y y, y, y. y y y y, y 7, y 7, y. y y y, y, y 7, y. y y y y 6, y, y, y. Prov h im-cling propry propry III, Appndix A. Hin: L u c, nd v / c, hn how L {f c } vu f u du. c. U propry IV of Appndix A o vrify h L {rcn / } in. 6. Apply propry V of Appndix A o f co b f n, n poiiv ingr 8 Zchry S Tng C- -
8 Zchry S Tng C- - Anwr C-.:. 6. 6.. 7 6. b 6 6., b 7. b b, b b b 8. 96 6 9. in co α β α β, b co in α β α β. 8. f co in. f 8 8. in 7 co f. f 7 6 6. f in 8 8 6. f co 7. f 8. in co f 9. f β α β α. y
. y. y 6 co in. y 6. y. y 6. y co in 7. y co in 8. y co in in 9. y. y. y 7. y. y co in 8 Zchry S Tng C- -