Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3% of h ol (Op book) Thi coi of o hour ppr wih 5 mulipl choic quio. () Fil Exm : 7% of h ol Th ppr coi of 6 quio d 5 quio hould b wrd. 3
. Lplc rform I mhmic, " rform" uully rfr o dvic which chg o kid of fucio or quio io ohr kid. O mp o dig rform which chg problm h w do o kow how o olv io problm which r o olv. Thi chiqu h provd vry ffciv i olvig diffril quio.my diffr rform hv b ivd. I hi cio w udy o of hm, h Lplc rform which r grlly pplid o complx lcricl circui d mchicl ym. Dfiiio Th Lplc rform L{ f ( )} of fucio f() i dfid o b L{ f ( )} f ( ) d F( ) whvr hi igrl xi. Th igrio vribl i. Hc h igrl dfi fucio of h w vribl. W hll lo cuomrily u h vribl of our origil fucio d h vribl of i Lplc rform. Dfiiio Th ivr Lplc rform of F() i fucio f() uch h L{f()} = F(). If w do h oprio of kig Lplc rform by L, d of kig ivr Lplc rform by L, h L{ f ( )} F( ) impli d covrly, L { F( )} f ( ) L { F( )} f ( ) impli L{ f ( )} F( ) Exrci Uig h dfiiio of Lplc rform how h followig. (i) L{} =, ( >) (iii) L Si, (ii) For >, L{Co()} = (iv) L, 4
No: I i how by rpd igrio h L{ } =! for y poiiv igr. Thorm : (i) L{f() g()} = L{f()} L{g()} whvr ll h rform xi. Hc L L f ( ) Lg ( ) f ( ) g( ) (ii) For y rl umbr,l{ f()} = L{f()} whvr boh id xi. L L f ( ) f (. Hc ) Exrci Uig bov horm fid (i) L Sih 5 (ii) L{4 Sih(3 ) 8 } (iii) L 3 8 Lplc rform of bic fucio r giv blow. f () L f ()! 3! Si Co (Abov formul r vlid for. ) Sih Coh ( Th bov hr formul r vlid for ). 5
Exrci 3 Uig h dfiiio of ivr Lplc rform obi h followig. 3 (i) L 5 (ii) L (iii) L 3 64 ( 4) (iv) L 3 5 { } (v) L { } 7 ( )( )( 4) Thorm : If i y rl umbr h L{ f()} = F(-) whr F() = L{f()} i.. L{ f()}= L{f()} Thi i kow fir rlio horm or Shifig propry. Exrci 4 Fid h followig. (i) L{ 5 3 } (iv) Cob (vii) Sih b (ii) L{ Co4} (iii) L { } 3 ( ) L (v) L Sib (vi) L Cohb L (viii) L CohCo b Lplc Trform of Driviv Our gol i o u h Lplc rform o olv cri kid of diffril dy d y quio. For h w d o vlu quii uch L{ } d L{ }. d d if f i coiuou h, for L{ f ()}= F() f() L{ f ()} F( ) f () f () Exrci 5 Prov bov wo rul. 6
No: I grl, L{ f ( )} whr F() = L{f()}. F( ) f () f ()... f ( ) Solvig diffril quio uig Lplc rform () I ordr o olv diffril quio Lplc rform of driviv r ud. Exrci 6 Solv h followig. (i) dy 3y ubjc o y() =. d 3 (ii) y 6y 9y ubjc o y() =, y ( ) 6 (iii) y 4 y 6y ubjc o y() =, y ( ) ( iv) x 6x Co4 ubjc o X() =, x ( ) Solvig Simulou diffril quio uig Lplc rform Th Lplc rform rduc ym of lir quio wih co coffici o of imulou lgbric quio i h rformd fucio. Exrci 7 Solv h followig. (i) x y y x y ubjc o x() =, y()= (ii)mchicl ym wih wo dgr of frdom ifi h quio miod blow. d x dy d y dx 3 4, 3. U lplc rform o drmi x d y d d d d dx dy y i giv h x, y,, ll vih =. d d 7
Som impor horm i lplc rform Thorm 3: L{ f ( u) du} F( ) whr L{f()} = F(). Thorm 4: If { f ( )} F( ) Similrly, d L{ f ( )} d I grl d d L h L{ f ( )} ( F( )). { F( )} d d L { f ( )} ( ) { F( )} Exrci 8 Fid h followig. (i) L{ Co } (iv) L ( ) (ii) L{ Si} (iii) L hc fid ( ) L ( ) Thorm 5: f ( ) L F( )d whr F( ) Lf ( ) h f ( ) L F( ) d. Exrci 9 Fid lplc rform of (i) Si u u du (ii) Si 3 Thorm 6: If f() i priodic wih priod T>, h T Lf ( ) ( f ( T )d ) 8
Exrci Fid L Si. Thorm 7: If U ( ) i dfid follow, ( U ),, Exrci L U ( ). h LU ( ) (i)obi h followig () If f() = k { U ( ) U b( )} h L f (). (b) If g () U( ) U ( ) U ( ) U ( )... h. L g() 3 (iii) Obi h grph of f () U ( ). (iv)for > kch h grph of h followig. () f ( ) U ( ) ( ) (b) f ( ) U ( ) ( (c) f ) U ( ) U ( ) (v) Expr h followig fucio i rm of ui p fucio., F() =, 3, 3 Hc fid lplc of F(). Thorm 8: Scod Shifig Propry If Lf ( ) F( ), L{ U ( ) f ( )} F( ). 9
Exrci : u f ( u) du F( ). Cd () A fucio f() i dfid by f ( ) 4, 3, Skch h grph of h fucio d drmi i Lplc rform. () Expr h followig i rm of ui p fucio. 8, () f ( ) 6, (b) f ( ) 3,, 3 (3) A ric R i ri wih iducc L i cocd wih E(). Th curr i i giv by, di L R i E( ) d If h wich i cocd = d dicocd =, fid h curr i() i rm of. 4) If f ( ) L ( )( 4 drmi f() d kch h grph of h fucio. Thorm 9: Covoluio Thorm If L { F( )} f ( ) d L { G( )} g( ) h L { F( ). G( )} f ( u) g( u) du
Exrci 4: () Applyig covoluio horm olv h followig iiil vlu problm. y y Si3, y ( ) d y ( ). ()Apply covoluio horm o fid h followig. () L ) (ii) ( ) L ( )( b )