The Laplace Transform

Transcription

1 7 he Laplace ranform 7 Definiion of he Laplace ranform 7 Invere ranform and ranform of Derivaive 7 Invere ranform 7 ranform of Derivaive 73 Operaional Properie I 73 ranlaion on he -Axi 73 ranlaion on he -Axi 74 Operaional Properie II 74 Derivaive of a ranform 74 ranform of Inegral 743 ranform of a Periodic Funcion 75 he Dirac Dela Funcion 76 Syem of Linear Differenial Equaion Chaper 7 in Review In he linear mahemaical model for a phyical yem uch a a pring/ma yem or a erie elecrical circui, he righ-hand memer, or inpu, of he differenial equaion m d x dx d d kx f() or Ld q Rdq d d q E() C i a driving funcion and repreen eiher an exernal force f() or an impreed volage E() In Secion 5 we conidered prolem in which he funcion f and E were coninuou However, diconinuou driving funcion are no uncommon For example, he impreed volage on a circui could e piecewie coninuou and periodic, uch a he awooh funcion hown on he lef Solving he differenial equaion of he circui in hi cae i difficul uing he echnique of Chaper 4 he Laplace ranform udied in hi chaper i an invaluale ool ha implifie he oluion of prolem uch a hee 73 Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i

2 74 CHAPER 7 HE LAPLACE RANSFORM 7 DEFINIION OF HE LAPLACE RANSFORM REVIEW MAERIAL Improper inegral wih infinie limi of inegraio Inegraion y par and parial fracion decompoiion INRODUCION In elemenary calculu you learned ha differeniaion and inegraion are ranform; hi mean, roughly peaking, ha hee operaion ranform a funcion ino anoher funcion For example, he funcion f(x) x i ranformed, in urn, ino a linear funcion and a family of cuic polynomial funcion y he operaion of differeniaion and inegraion: d dx x x and x dx 3 x3 c Moreover, hee wo ranform poe he lineariy propery ha he ranform of a linear cominaion of funcion i a linear cominaion of he ranform For a and conan d dx [ f (x) g(x)] f (x) g(x) and [f (x) g(x)] dx f (x) dx g(x) dx provided ha each derivaive and inegral exi In hi ecion we will examine a pecial ype of inegral ranform called he Laplace ranform In addiion o poeing he lineariy propery he Laplace ranform ha many oher inereing properie ha make i very ueful in olving linear iniial-value prolem Inegral ranform If f(x, y) i a funcion of wo variale, hen a definie inegral of f wih repec o one of he variale lead o a funcion of he oher variale For example, y holding y conan, we ee ha xy dx 3y Similarly, a defi nie inegral uch a a K(, ) f () d ranform a funcion f of he variale ino a funcion F of he variale We are paricularly inereed in an inegral ranform, where he inerval of inegraion i he unounded inerval [, ) If f () i defined for, hen he improper inegral K(, ) f () d i defined a a limi K(, ) f () d lim K(, ) f () d () : We will aume hroughou ha i a real variale If he limi in () exi, hen we ay ha he inegral exi or i convergen; if he limi doe no exi, he inegral doe no exi and i divergen he limi in () will, in general, exi for only cerain value of he variale A Definiio he funcion K(, ) in () i called he kernel of he ranform he choice K(, ) e a he kernel give u an epecially imporan inegral ranform DEFINIION 7 Laplace ranform Le f e a funcion defined for hen he inegral { f ()} e f () d i aid o e he Laplace ranform of f, provided ha he inegral converge he Laplace ranform i named in honor of he French mahemaician and aronomer Pierre-Simon Marqui de Laplace (749 87) () Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i

3 7 DEFINIION OF HE LAPLACE RANSFORM 75 When he defining inegral () converge, he reul i a funcion of In general dicuion we hall ue a lowercae leer o denoe he funcion eing ranformed and he correponding capial leer o denoe i Laplace ranform for example, {f ()} F(), {g()} G(), {y()} Y() A he nex four example how, he domain of he funcion F() depend on he funcion f() EXAMPLE Applying Definiion 7 Evaluae {} SOLUION From (), {} e () d lim e lim : provided ha In oher word, when, he exponen i negaive, and e : a : he inegral diverge for he ue of he limi ign ecome omewha ediou, o we hall adop he noaion a a horhand for wriing lim : () For example, {} e () d e, : e d lim : e A he upper limi, i i underood ha we mean e : a : for EXAMPLE Applying Definiion 7 Evaluae {} SOLUION From Definiion 7 we have {} e d Inegraing y par and uing lim, along wih he reul from Example, we oain : e, {} e e d {} EXAMPLE 3 Applying Definiion 7 Evaluae (a) {e 3 } () {e 5 } SOLUION In each cae we ue Definiion 7 (a) {e 3 } e 3 e d e (3) d e(3) 3 3 he la reul i valid for 3 ecaue in order o have lim : e (3) we mu require ha 3 or 3 Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i

4 76 CHAPER 7 HE LAPLACE RANSFORM () {e 5 } e 5 e d e (5) d e(5) 5 5 In conra o par (a), hi reul i valid for 5 ecaue lim : e (5) demand 5 or 5 EXAMPLE 4 Applying Definiion 7 Evaluae {in } SOLUION oain From Definiion 7 and wo applicaion of inegraion y par we {in } e in d A hi poin we have an equaion wih {in } on oh ide of he equaliy Solving for ha quaniy yield he reul {in } 4, I a Linear ranform For a linear cominaion of funcion we can wrie e [ f () g()] d e f () d e g() d whenever oh inegral converge for c Hence i follow ha { e in e co d, e co d lim e co, Laplace ranform of in : e co [ e in d] 4 {in } f () g()} { f ()} {g()} F() G() (3) Becaue of he propery given in (3), i aid o e a linear ranform EXAMPLE 5 Lineariy of he Laplace ranform In hi example we ue he reul of he preceding example o illurae he lineariy of he Laplace ranform (a) From Example and we have for, { 5} {} 5 {} 5 () From Example 3 and 4 we have for 5, {4e 5 in } 4 {e 5 } {in} Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i

5 7 DEFINIION OF HE LAPLACE RANSFORM 77 (c) From Example,, and 3 we have for, {e 3 7 9} {e 3 } 7 {} 9 {} We ae he generalizaion of ome of he preceding example y mean of he nex heorem From hi poin on we hall alo refrain from aing any rericion on ; i i underood ha i ufficienly rericed o guaranee he convergence of he appropriae Laplace ranform HEOREM 7 ranform of Some Baic Funcion (a) {} () (d) (f) { n } n! n, n,, 3, {in k} k k {inh k} k k (c) (e) (g) {e a } a {co k} k {coh k} k hi reul in () of heorem 7 can e formally juified for n a poiive ineger uing inergraion y par o fir how ha { n } n { n } hen for n,, and 3, we have, repecively, {} {} f() a FIGURE 7 funcion 3 Piecewie coninuou { } {} 3 { 3 } 3 { } If we carry on in hi manner, you hould e convinced ha { n } n 3 n n! n Sufficien Condiion for Exience of {f()} he inegral ha define he Laplace ranform doe no have o converge For example, neiher {>} nor {e } exi Sufficien condiion guaraneeing he exience of {f ()} are ha f e piecewie coninuou on [, ) and ha f e of exponenial order for Recall ha a funcion f i piecewie coninuou on [, ) if, in any inerval a, here are a mo a finie numer of poin k, k,,, n ( k k ) a which f ha finie diconinuiie and i coninuou on each open inerval ( k, k ) See Figure 7 he concep of exponenial order i defined in he following manne DEFINIION 7 Exponenial Order A funcion f i aid o e of exponenial order if here exi conan c, M, and uch ha f () Me c for all Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i

6 78 CHAPER 7 HE LAPLACE RANSFORM f() FIGURE 7 order Me c ( c > ) f ( ) f i of exponenial If f i an increaing funcion, hen he condiion f() Me c,, imply ae ha he graph of f on he inerval (, ) doe no grow faer han he graph of he exponenial funcion Me c, where c i a poiive conan See Figure 7 he funcion f (), f () e, and f () co are all of exponenial order ecaue for c, M, we have, repecively, for e, e e, and co e A comparion of he graph on he inerval [, ) i given in Figure 73 f() e f() e f() e co e (a) () (c) FIGURE 73 hree funcion of exponenial order A poiive inegral power of i alway of exponenial order, ince, for c, n Me c or n e M for c f() e e c i equivalen o howing ha lim : n >e c i finie for n,, 3, he reul follow from n applicaion of L Hôpial rule A funcion uch a f() e i no of exponenial order ince, a hown in Figure 74, e grow faer han any poiive linear power of e for > c > hi can alo e een from FIGURE 74 order c e i no of exponenial a : HEOREM 7 e e c e c e (c) : Sufficien Condiion fo Exience If f i piecewie coninuou on [, ) and of exponenial order, hen exi for c { f ()} PROOF By he addiive inerval propery of definie inegral we can wri { f()} e f() d e f() d I I he inegral I exi ecaue i can e wrien a a um of inegral over inerval on which e f () i coninuou Now ince f i of exponenial order, here exi conan c, M, o ha f () Me c for We can hen wrie I e f () d M e e c d M e (c) d M e(c) c for c Since Me (c) d converge, he inegral e f () d converge y he comparion e for improper inegral hi, in urn, implie ha I exi Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i

7 7 DEFINIION OF HE LAPLACE RANSFORM 79 for c he exience of I and I implie ha {f ()} e f () d exi for c EXAMPLE 6 ranform of a Piecewie Coninuou Funcion Evaluae {f()} where f (), 3, 3 y 3 FIGURE 75 Piecewie coninuou funcion in Example 6 SOLUION he funcion f, hown in Figure 75, i piecewie coninuou and of exponenial order for Since f i defined in wo piece, {f()} i expreed a he um of wo inegral: {f ()} e f () d 3 e () d e () d e 3 e3, 3 We conclude hi ecion wih an addiional i of heory relaed o he ype of funcion of ha we will, generally, e working wih he nex heorem indicae ha no every arirary funcion of i a Laplace ranform of a piecewie coninuou funcion of exponenial order HEOREM 73 Behavior of F() a : If f i piecewie coninuou on [, ) and of exponenial order and F() {f()}, hen lim : F() PROOF Since f i of exponenial order, here exi conan g, M, and o ha f() M e g for Alo, ince f i piecewie coninuou for, i i necearily ounded on he inerval; ha i, f() M M e If M denoe he maximum of he e {M, M } and c denoe he maximum of {, g}, hen F() e f () d M e e c d M e (c) d M c for c A :, we have F() :, and o F() { f ()} : REMARKS (i) hroughou hi chaper we hall e concerned primarily wih funcion ha are oh piecewie coninuou and of exponenial order We noe, however, ha hee wo condiion are ufficien u no neceary for he exience of a Laplace ranform he funcion f () / i no piecewie coninuou on he inerval [, ), u i Laplace ranform exi he funcion f() e co e i no of exponenial order, u i can e hown ha i Laplace ranform exi See Prolem 43 and 54 in Exercie 7 (ii) A a conequence of heorem 73 we can ay ha funcion of uch a F () and F () ( ) are no he Laplace ranform of piecewie coninuou funcion of exponenial order, ince F () :/ and F () :/ a : Bu you hould no conclude from hi ha F () and F () are no Laplace ranform here are oher kind of funcion Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i

8 8 CHAPER 7 HE LAPLACE RANSFORM EXERCISES 7 In Prolem 8 ue Definiion 7 o find {f()} f (),, f () 4,, f (),, f (),, f () in,, f (), co, f() > > (, ) FIGURE 76 Graph for Prolem 7 FIGURE 77 Graph for Prolem 8 FIGURE 78 Graph for Prolem 9 FIGURE 79 Graph for Prolem f() f() f() c a (, ) f() e 7 f() e 5 3 f() e 4 4 f() e 5 f() e in 6 f() e co 7 f() co 8 f() in In Prolem 9 36 ue heorem 7 o find { f ()} 9 f() 4 f() 5 f() 4 f() 7 3 Anwer o eleced odd-numered prolem egin on page ANS- 3 f() f() f() ( ) 3 6 f() ( ) 3 7 f() e 4 8 f() e f() ( e ) 3 f() (e e ) 3 f() 4 5 in 3 3 f() co 5 in 33 f() inh k 34 f() coh k 35 f() e inh 36 f() e coh In Prolem 37 4 find {f()} y fir uing a rigonomeric ideniy 37 f() in co 38 f() co 39 f() in(4 5) 4 4 We have encounered he gamma funcion (a) in our udy of Beel funcion in Secion 64 (page 58) One definiion of hi funcion i given y he improper inegral () a e d, a Ue hi definiion o how ha (a ) a(a) 4 Ue Prolem 4 and a change of variale o oain he generalizaion {} ( ), of he reul in heorem 7() f () co 6, In Prolem ue Prolem 4 and 4 and he fac ha ( ) p o find he Laplace ranform of he given funcion 43 f() / 44 f() / 45 f() 3/ 46 f() / 8 5/ Dicuion Prolem 47 Make up a funcion F() ha i of exponenial order u where f() F() i no of exponenial order Make up a funcion f ha i no of exponenial order u whoe Laplace ranform exi 48 Suppoe ha {f ()} F () for c and ha {f ()} F () for c When doe {f () f ()} F () F ()? 49 Figure 74 ugge, u doe no prove, ha he funcion f () e i no of exponenial order How doe he oervaion ha ln M c, for M and ufficienly la ge, how ha e Me c for any c? 5 Ue par (c) of heorem 7 o how ha {e (ai) a i }, where a and are real ( a) Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i

9 7 INVERSE RANSFORMS AND RANSFORMS OF DERIVAIVES 8 and i Show how Euler formula (page 33) can hen e ued o deduce he reul a ( a) {e a co } {e a in } ( a) 5 Under wha condiion i a linear funcion f (x) mx, m, a linear ranform? 5 Explain why he funcion, f() 4, 5 >( 5), 5 i no piecewie coninuou on [, ) 53 Show ha he funcion f() > doe no poe a Laplace ranform [Hin: Wrie a wo improper inegral: e e d d I I {> } Show ha I diverge] {> } 54 Show ha he Laplace ranform exi [Hin: Sar wih inegraion y par] 55 If and a i a conan, how ha hi reul i known a he change of cale heor em 56 Ue he given Laplace ranform and he reul in Prolem 55 o find he indicaed Laplace ranform Aume ha a and k are poiive conan (a) () (c) {f()} F() {e } ; {in } { co } {f(a)} a F a {e a } ; ( ) ; (d) {in inh } 4 4 ; {in k} {e coe } { co k} {in k inh k} 7 INVERSE RANSFORMS AND RANSFORMS OF DERIVAIVES REVIEW MAERIAL Parial fracion decompoiion See he Suden Reource Manual INRODUCION In hi ecion we ake a few mall ep ino an inveigaion of how he Laplace ranform can e ued o olve cerain ype of equaion for an unknown funcion We egin he dicuion wih he concep of he invere Laplace ranform or, more preciely, he invere of a Laplace ranform F() Afer ome imporan preliminary ackground maerial on he Laplace ranform of derivaive f(), f(),, we hen illurae how oh he Laplace ranform and he invere Laplace ranform come ino play in olving ome imple ordinary differenial equaion 7 INVERSE RANSFORMS he Invere Prolem If F() repreen he Laplace ranform of a funcion f(), ha i,, we hen ay f () i he invere Laplace ranform of F() and wrie f() {F()} For example, from Example,, and 3 of Secion 7 we have, repecively, {f()} F() ranform {} {} {e 3 } 3 Invere ranform e 3 3 Copyrigh Cengage Learning All Righ Reerved May no e copied, canned, or duplicaed, in whole or in par Due o elecronic righ, ome hird pary conen may e uppreed from he ebook and/or echaper() Ediorial review ha deemed ha any uppreed conen doe no maerially affec he overall learning experience Cengage Learning reerve he righ o remove addiional conen a any ime if uequen righ rericion require i