6.8 Laplace Transform: General Formulas

Transcription

1 48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy 6. l{e a f ()} F( a) l {F( a)} e a f () -Shifing (Fir Shifing Theorem) 6. l( f r) l( f ) f () l( f ) l( f ) f () f r() l( f (n) ) n l( f ) (n ) f () Á Á f (n ) () l e f () df l( f ) Differeniaion of Funcion Inegraion of Funcion 6. ( f * g)() f ()g( ) d f ( )g() d l( f * g) l( f )l(g) onvoluion 6.5 l{ f ( a) u( a)} e a F() l {e a F ()} f ( a) u( a) -Shifing (Second Shifing Theorem) 6. l{f ()} Fr() l e f () f F( ) d Differeniaion of Tranform Inegraion of Tranform 6.6 l( f ) p p e f () d e f Periodic wih Period p 6.4 Projec 6

2 SE. 6.9 Table of Laplace Tranform Table of Laplace Tranform For more exenive able, ee Ref. [A9] in Appendix. F () l{ f ()} > > > n > > > > a (n,, Á ) (a ) n >(n )! > p >p a > (a) f () Sec a ( a) ( a) n (n,, Á ) ( a) k (k ) e a e a (n )! n e a (k) k e a 6. ( a)( b) ( a)( b) (a b) (a b) a b (ea e b ) a b (aea be b ) v v a a ( a) v a ( a) v in v v co v inh a a coh a v ea inh v e a co v 6. 9 ( v ) ( v ) ( co v) v (v in v) v x 6. (coninued )

3 5 HAP. 6 Laplace Tranform Table of Laplace Tranform (coninued ) F () l{ f ()} f () Sec. 4 ( v ) ( v ) ( v ) ( a )( b ) (a b ) (in v v co v) v in v v (in v v co v) v (co a co b) b a k 4 4 4k 4 4 k 4 4 k 4 (in k co k co k inh k) 4k in k inh k k (inh k in k) k (coh k co k) k 9 e (a b)> I a a b b I 5.5 a b J (a) J 5.4 a p ea ( a) ( a) > p (k) a k > (k ) a b I k > (a) ( a ) k I e a > u( a) 6. 5 e a d( a) J (k) J a b e k> e k> ek> > e k (k ) p (eb e a ) co k p inh k pk k p e k >4 (coninued )

4 haper 6 Review Queion and Problem 5 Table of Laplace Tranform (coninued ) F () l{ f ()} f () Sec. 4 ln ln g (g.577) g ln a b (eb e a ) 4 ln v ( co v) ln a arcan v arcco ( coh a) in v Si() App. A. HAPTER 6 REVIEW QUESTIONS AND PROBLEMS. Sae he Laplace ranform of a few imple funcion from memory.. Wha are he ep of olving an ODE by he Laplace ranform?. In wha cae of olving ODE i he preen mehod preferable o ha in hap.? 4. Wha propery of he Laplace ranform i crucial in olving ODE? 5. I l{f () g()} l{f ()} l{g()}? l{f ()g()} l{f ()}l{g()}? Explain. 6. When and how do you ue he uni ep funcion and Dirac dela? 7. If you know f () l {F()}, how would you find l {F()> }? 8. Explain he ue of he wo hifing heorem from memory. 9. an a diconinuou funcion have a Laplace ranform? Give reaon.. If wo differen coninuou funcion have ranform, he laer are differen. Why i hi pracically imporan? 9 LAPLAE TRANSFORMS Find he ranform, indicaing he mehod ued and howing he deail.. 5 coh inh. e (co 4 in 4). in ( p) 4. 6 u( 4) 5. e > u( ) 6. u( p) in 7. co in 8. (in v) * (co v) 9. * e 8 INVERSE LAPLAE TRANSFORM Find he invere ranform, indicaing he mehod ued and howing he deail: e v co u in u v ( ) ( 6.5) 4 6. e ODE AND SYSTEMS Solve by he Laplace ranform, howing he deail and graphing he oluion: 9. y 4yr 5y 5, y() 5, yr() 5. y 6y 4d( p), y(), yr()

5 5 HAP. 6 Laplace Tranform. y yr y u( p) in, y(), yr(). y 4y d( p) d( p), y(), yr() y yr y u( ), y(), yr(). 4. y r y, y r 4y d( p), y (), y () 5. y r y 4y, y () y r y y, y (), 6. y r y 4y, y r y y, y () 4, y () 4 7. y r y u( p), y r y u( p), y (), y () 8 45 MASS SPRING SYSTEMS, IRUITS, NETWORKS Model and olve by he Laplace ranform: 8. Show ha he model of he mechanical yem in Fig. 49 (no fricion, no damping) i m y m y k y k ( y y ) k ( y y ) k y ). y k k k y 4. Find and graph he charge q() and he curren i() in he L-circui in Fig. 5, auming L H, F, v() e if p, v() if p, and zero iniial curren and charge. 4. Find he curren i() in he RL-circui in Fig. 5, where R 6, L H,. F, v() 7 in V, and curren and charge a are zero. v() v() Fig. 5. L-circui Fig. 5. RL-circui 44. Show ha, by Kirchhoff Volage Law (Sec..9), he curren in he nework in Fig. 5 are obained from he yem Li r R(i i ) v() R(i r i r) i. Solve hi yem, auming ha R, L H,.5 F, v V, i (), i () A. L L R L Fig. 49. Syem in Problem 8 and 9 v() i i R 9. In Prob. 8, le m kg>ec m kg, k k, k 4 kg>ec. Find he oluion aifying he iniial condiion y () y (), y r() meer>ec, y r() meer>ec. 4. Find he model (he yem of ODE) in Prob. 8 exended by adding anoher ma m and anoher pring of modulu k 4 in erie. 4. Find he curren i() in he R-circui in Fig. 5, where R,. F, v() V if 4, v() 4 V if 4, and he iniial charge on he capacior i. Fig. 5. Nework in Problem Se up he model of he nework in Fig. 54 and find he oluion, auming ha all charge and curren are when he wich i cloed a. Find he limi of i () and i () a :, (i) from he oluion, (ii) direcly from he given nework. L = 5 H i i V =.5 F R Swich Fig. 54. Nework in Problem 45 v() Fig. 5. R-circui

6 Summary of haper 6 5 SUMMARY OF HAPTER 6 Laplace Tranform The main purpoe of Laplace ranform i he oluion of differenial equaion and yem of uch equaion, a well a correponding iniial value problem. The Laplace ranform F() l( f ) of a funcion f () i defined by () F() l( f ) e f () d (Sec. 6.). Thi definiion i moivaed by he propery ha he differeniaion of f wih repec o correpond o he muliplicaion of he ranform F by ; more preciely, l( f r) l( f ) f () () (Sec. 6.) l( f ) l( f ) f () f r() ec. Hence by aking he ranform of a given differenial equaion () y ayr by r() (a, b conan) and wriing l(y) Y(), we obain he ubidiary equaion (4) ( a b)y l(r) f () f r() af (). Here, in obaining he ranform l(r) we can ge help from he mall able in Sec. 6. or he larger able in Sec Thi i he fir ep. In he econd ep we olve he ubidiary equaion algebraically for Y(). In he hird ep we deermine he invere ranform y() l (Y), ha i, he oluion of he problem. Thi i generally he harde ep, and in i we may again ue one of hoe wo able. Y() will ofen be a raional funcion, o ha we can obain he invere l (Y) by parial fracion reducion (Sec. 6.4) if we ee no impler way. The Laplace mehod avoid he deerminaion of a general oluion of he homogeneou ODE, and we alo need no deermine value of arbirary conan in a general oluion from iniial condiion; inead, we can iner he laer direcly ino (4). Two furher fac accoun for he pracical imporance of he Laplace ranform. Fir, i ha ome baic properie and reuling echnique ha implify he deerminaion of ranform and invere. The mo imporan of hee properie are lied in Sec. 6.8, ogeher wih reference o he correponding ecion. More on he ue of uni ep funcion and Dirac dela can be found in Sec. 6. and 6.4, and more on convoluion in Sec Second, due o hee properie, he preen mehod i paricularly uiable for handling righ ide r() given by differen expreion over differen inerval of ime, for inance, when r() i a quare wave or an impule or of a form uch a r() co if 4p and elewhere. The applicaion of he Laplace ranform o yem of ODE i hown in Sec (The applicaion o PDE follow in Sec...)