Vidyalankar. Vidyalankar S.E. Sem. III [BIOM] Applied Mathematics - III Prelim Question Paper Solution. 1 e = 1 1. f(t) =

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1 . (a). (b). (c) f() L L e i e Vidyalakar S.E. Sem. III [BIOM] Applied Mahemaic - III Prelim Queio Paper Soluio L el e () i ( ) H( ) u e co y + 3 3y u e co y + 6 uy e i y 6y uyy e co y 6 u + u yy e co y + 6 e co y 6 u aifie Laplace equaio u i Harmoic. If i a eve ieger he ice, m (m + )!, we have J () ( ) m ( / ) m m m!(m )! m m ( ) ( ) J () m mm!(m )! If i eve, (m + ) i eve & () m+ m+ J () ( ) m ( / ) m J () m m!(m )! Hece, J () i eve. Agai, if i odd he (m + ) i odd & hece, () m+ ( m ) Vidyalakar J () Hece, J () i odd ( ) m ( ) ( / ) m m m!(m )! ( ) m ( / ) m m m!(m )! J () 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol

2 Vidyalakar : S.E. Mah III. (d) We have, u a + b y + cy + d y u a3 +by + d u y b y + cy 3 y v 3 y ey 3 + y v y ey 3 + y v y 3 3ey + f(z) i aalyic. I aifie auchy Riema equaio. u v y a 3 + by + d 3 3ey + omparig boh ide, we ge a a b 3e d d Alo u y v b y + cy 3 y y + ey 3 y omparig boh ide, we ge b b 6 c e From (), we ge b 3e 3e e From (), we have c e c c a, b 6, c, d, e a a f() d d f() co d co d () (). (a) Vidyalakar 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol

3 Prelim Queio Paper Soluio i co ( ) ( ), ifiodd, if ieve The Half Rage coie erie i give by a f() a co ( ) co. (b) The curve i he boudary of he riagle i oy plae a z. F y î ˆj (z) k ˆ where z y î ˆj k ˆ î ˆj kˆ F y z A y î ˆj ( ) ( y) k ˆ ˆj(y) k ˆ ( F ). ˆ ˆ j ( y) k ˆ. ˆk a ˆ ˆk. ( y) We have by Soke heorem. F. dr ( F). d ˆ S S ( ) ( y) d ( y) d dy y ( y) dy d y d R (,, ) z O (,, ) y O (, ) (,, ) B B (, ) Vidyalakar y A (, ) y 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol 3

4 Vidyalakar : S.E. Mah III. () (i) d LH e 3 d 3 LiH e Li e Lco H( a) e a a Lf()H( a) e Lf( a) e 3 3 LiH H LiH LH e 3 e d (ii) Sice, J J + () d d J d d J d + J () J d + () Pu i equaio (), J3 d J () + () Puig i equaio (), J d J () + (3) Now, J 3 d ( ) J () + J d + J J 3 d J () + Vidyalakar 3. (a) Fid he Fourier erie for f() i (, ). Hece deduce ha a f() d d d 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol

5 Prelim Queio Paper Soluio 3. (b) a f() co d co d i co () ( ) f() i eve fucio b. a f() aco co d co ( ) ( ) co co co 3 co co co3 co Puig, we ge re coi e ij i.e. e co ; y e i d e (co i ) d, dy e (i + co ) d dr [e (coi) i e ico) j ] d + y e (co + i ) e ( + y ) 3/ (e ) 3/ e 3 F.dr e(coi ij). e [ (co i ) i + (i + co ) j ] d 3 e e 3 {co (co i ) i (i co )} d e e d - Vidyalakar e F.dr e d e e e F.dr e 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol 5

6 Vidyalakar : S.E. Mah III 3. (c) a f () aco bi a f()d π π a π π f() co d π - π π πd π π π π d co d i i co co () b π f() i d π - π i d co co i () co d i d co co co co co Vidyalakar Puig hee value i (), we ge f () co co co3 co i 3 i i3... i () 6 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol

7 Prelim Queio Paper Soluio. (a) Now f() i dicoiuou a. A a poi of dicoiuiy c f () lim f() lim f() c c f () Hece ubiuig i equaio (), we ge L{i 3u} 9 L{u i 3u} d 3 d 9 3() Lui u du Le ui3udu L f() Lf() 6 (ii) L{co } L{} L{co } co L 85 d d d () L f(u)du Vidyalakar f() L log log ()d log log 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol 7

8 Vidyalakar : S.E. Mah III. (b). (c) log log log log Le u 3 y + y 3 y u 6y + u y 3 3y y We kow ha dv v v u u d dy d dy y y 3 3y y d 6y dy 3 3y y d 6y dy Iegraig, we ge v 3 + 3y + y log orhogoal rajecory of family of curve i give by 3 + 3y + y 3 3y y whe (i) We kow ha +iy z () iy z () From () ad () we ge i z z, y (zz) y i z z y i z z We kow ha z y i z y z y Vidyalakar i (3) y z y i z y z y i () y 8 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol

9 Prelim Queio Paper Soluio 5. (a) From (3) ad () we ge zz z z i i y y i i y y y y z z y (ii) Noe ha, log f'(z) log f' (z) log [f'(z). f'(z) logf'(z) log [f'(z)] L.H.S. log f'(z) y By Gree Thm. Now pdqdy R logf'(z) logf'(z) z z logf'(z) logf'(z) z z z z () () ( z, z are idepede) z z Q P d dy y f dr yi y j didy ydy dy By compario P y, Q y Q y, P y f dr y ddy R pu r co, y r i, d dy r dr d ra(co ) r rdrd r y r a(+co) Vidyalakar 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol 9

10 Vidyalakar : S.E. Mah III 5. (b) Pu a( co ) r d a co d a ( co ) d 8 8a co d, d d & uig reducio formula, f dr 8a / 8 co d a a 6 5 z Sice w z 3 We have wz 3w 5 z z(w + ) 5 + 3w z 5 3w (w ) whe z, 5 3w (w ) 53(uiv) (u iv ) 53ui3v uiv 5 3u 9v 6 u v 5 + 3u + 9u + 9v 6(u + u + + v ) 6u 9u + 3u 3u v 9v 7u + u 9 + 7v 7u + u + 7v + 9 u uv u u v u v cere,, radiu Vidyalakar 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol

11 Prelim Queio Paper Soluio 5. (c) 6. (a) i i roo of J () Le he Beel fourier erie be, f() ij ( i) Muliplyig by J () o boh ide ad iegraig from o. J () i J ( ) J ( i) d i J ( ) 3 J( )d J( ) d d J () J () J () 3 J( )d J ()d 3 J ( ) J( ) J( ) J( ) i J 3 ( i) J( ) oider L f() i i i J( ) f () co co 3 3 d J ( ) 3 log ( ) log ( 3 ) log 3 log () By defiiio of Laplace Traform The equaio () mea coco3 3 e d log Puig co co d log log Vidyalakar 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol

12 Vidyalakar : S.E. Mah III 6. (b) 6. (c) 3z + + i Le u + iv ad z + iy u + iv 3 ( + iy) + + i 3 + 3iy + + i u + iv (3 + ) + i (3y + ) u 3 + ad v 3y + Now z k i.e. y k + y k Thi i a circle wih cere (, ) ad radiu k. k y k if y (k, ) if y k (, k) u 3k + ad v 3k + ad u ad v u 3k + ad v ad u ad v 3k + are he image of (k, ) ad (, k) for z k. a a b f() d 3 3 d 3 f() co d co d 6 3 ( ) 3 i co i ( )( ) 3 co [ co ] Vidyalakar f() i d ( ) id co i co ( )( ) 3 co co 3 3 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol

13 Prelim Queio Paper Soluio b 3 3 a Now, f() acob i z Pu, we ge co co co co Vidyalakar 3/Egg/SE/Pre Pap/3/BIOM/Mah III_Sol 3

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