Engineering Mathematics 2018

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1 Engineering Mahemaics 08 SUBJET NAME : Mahemaics II SUBJET ODE : MA65 MATERIAL NAME : Par A quesions REGULATION : R03 UPDATED ON : November 06 TEXTBOOK FOR REFERENE To buy he book visi : Sri Hariganesh Publicaions (Auhor:. Ganesan) (Scan he above Q.R code for he direc download of his maerial) Uni I (Vecor alculus). Find he value of m so ha he vecor 3 F x y i y j x m k is solenoidal. (Tex Book Page No.:.9). Find such ha F (3x y ) i ( x y ) j ( x y ) k is solenoidal. Tex Book Page No.:.6 3. Find he values of a, b, c so ha he vecor (Tex Book Page No.:.6) F x y a i bx y j. Find he direcional derivaive of x cy k may be irroaional. ( x, y, ) xy y a he poin,, in he direcion of he vecor i j 3k. (Tex Book Page No.:.8) 5. Find he direcional derivaive of xy a,, in he direcion of i j k. Tex Book Page No.:.6 6. Is he posiion vecor r xi yj k irroaional? Jusify. (Tex Book Page No.:.9) 7. Find curlf if F xyi yj xk. 8. Prove ha F yi xj xyk is irroaional. (Tex Book Page No.:.5) n 9. Find grad r where r xi yj k and r r. (Tex Book Page No.:.3) 0. Evaluae log r. Sri Hariganesh Publicaions (Ph: , ) Page

2 Engineering Mahemaics 08. Find he uni normal o he surface x xy a,,. (Tex Book Page No.:.). Find he uni normal vecor o he surface a,,5 x y.(tex Book Page No.:.8) 3. Prove ha div r 3and curl r 0. (Tex Book Page No.:.). Prove ha curl grad 0. (Tex Book Page No.:.0) 5. Sae Soke s heorem. (Tex Book Page No.:.0) 6. Sae Green s heorem. (Tex Book Page No.:.7) 7. Sae Gauss divergence heorem. (Tex Book Page No.:.93) 8. Prove by Green s heorem ha he area bounded by a simple closed curve is xdy ydx. (Tex Book Page No.:.90) Uni II (Ordinary Differenial Equaions). Solve d y dy y 0. dx dx D D y 0.. Solve 3. Solve he equaion 3. Solve D 6D 3 y 0. (Tex Book Page No.:.8) D D D y 0. D 3D 3D y Solve 6. Find he paricular inegral of he equaion 9 7. Solve D 3x D y e. y. (Tex Book Page No.:.55) 8. If i, i are he roos of he auxiliary equaion corresponding o a fourh order homogenous linear differenial equaion F( D) y 0, find is soluion. 9. Find he paricular inegral of D y cosh x. (Tex Book Page No.:.55) Sri Hariganesh Publicaions (Ph: , ) Page

3 Engineering Mahemaics Find he paricular inegral of. Find he paricular inegral of D y sin x.. Find he paricular inegral of 3. Find he paricular inegral of. Find he paricular inegral of D y sin x. (Tex Book Page No.:.56) x D D y e cos x. 5. Find he paricular inegral of 6. Solve he equaion x y xy y 0. x D y e cos x. (Tex Book Page No.:.0) x D D y e sin x. x D D y x e. 7. Solve d y dy x x y 0. (Tex Book Page No.:.59) dx dx 8. Transform he equaion coefficiens. 9. onver ino a linear differenial equaion wih consan x y xy x (Tex Book Page No.:.58) 3x D 5xD 7 y / x log x ino an equaion wih consan coefficiens. Tex Book Page No.:.9 0. Transform he equaion equaion wih consan coefficiens. Uni III (Laplace Transform) d y dy (x 3) (x 3) y 6x dx dx ino a differenial. Sae he condiions under which Laplace ransform of f() exiss. (Tex Book Page No.:3.). Find he Laplace ransform of uni sep funcion. (Tex Book Page No.:3.9) 3. Sae he firs shifing heorem on Laplace ransforms. (Tex Book Page No.:3.9). Evaluae e sin d using Laplace ransform Find Le 3 sin cos. (Tex Book Page No.:3.) 6. Find he Laplace ransform of e sin. (Tex Book Page No.:3.) Sri Hariganesh Publicaions (Ph: , ) Page 3

4 Engineering Mahemaics Find he Laplace ransform of e. (Tex Book Page No.:3.3) 8. cos Find he Laplace ransform of. (Tex Book Page No.:3.8) 9. Find sin L. (Tex Book Page No.:3.9) 0. Find Laplace ransform of sin. (Tex Book Page No.:3.5). Sae iniial and final value heorem. (Tex Book Page No.:3.36). Sae convoluion heorem on Laplace ransforms. (Tex Book Page No.:3.99) 3. Verify he final value heorem for f ( ) 3e. (Tex Book Page No.:3.30). Verify iniial value heorem for he funcion f () ae b. (Tex Book Page No.:3.38) 5. Find 6. Find L L s s s 6s3. (Tex Book Page No.:3.6). (Tex Book Page No.:3.63) 7. Find he inverse Laplace ransform of s s. (Tex Book Page No.:3.65) 8. Find f() if he Laplace ransform of () s f is s. (Tex Book Page No.:3.63) 9. Find L co ( s). (Tex Book Page No.:3.88) 0. Find L log s s a. Uni IV (Analyic Funcions). Verify f () 3 is analyic or no. (Tex Book Page No.:.0). Show ha is no analyic a any poin. (Tex Book Page No.:.) Sri Hariganesh Publicaions (Ph: , ) Page

5 Engineering Mahemaics Define harmonic funcion. (Tex Book Page No.:.). Show ha u x x xy 3 3 is harmonic. (Tex Book Page No.:.35) 5. Verify wheher he funcion Tex Book Page No.:.37 u x xy x y is harmonic. 6. Define onformal mapping. (Tex Book Page No.:.59) 7. Find he map of he circle 3 under he ransformaion w. (Tex Book Page No.:.59) 8. Find he image of he line x k under he ransformaion w. (Tex Book Page No.:.6) 9. Sae he auchy-riemann equaion in polar coordinaes saisfied by an analyic funcion. Tex Book Page No.:.7 0. Prove ha a bilinear ransformaion has a mos wo fixed poins. (Tex Book Page No.:.8). Find he fixed poins of mapping w 6 9. (Tex Book Page No.:.83). Find he invarian poins of he ransformaion w 6 7. (Tex Book Page No.:.83) 3. Find he invarian poins of he ransformaion w. (Tex Book Page No.:.8). Find he invarian poins of a funcion 3 7 f( ). 7 6i 5. Find he invarian poins of f (). (Tex Book Page No.:.8) 6. Find he criical poins of he ransformaion 7. Find he criical poins of he ransformaion w. (Tex Book Page No.:.78) w ( )( ). (Tex Book Page No.:.79) 8. Find he consans ab, if f ( ) x ay i(3 x by) is analyic. (Tex Book Page No.:.8) 9. Verify wheher f () is analyic funcion or no. (Tex Book Page No.:.8) Sri Hariganesh Publicaions (Ph: , ) Page 5

6 Engineering Mahemaics Are,Re( ),Im( ) analyic? Give reason. (Tex Book Page No.:.8) Uni V (omplex Inegraion). Define Singular poin.. Define and give an example of essenial singular poins. (Tex Book Page No.:5.65) 3. Expand f( ) as a Taylor series abou he poin.. Expand f ( ) sin in a Taylor series abou origin. (Tex Book Page No.:5.63) 5. Evaluae an d where is. (Tex Book Page No.:5.3) c 6. Find he Taylor series for f ( ) sin abou. (Tex Book Page No.:5.38) 7. Sae auchy s inegral heorem. (Tex Book Page No.:5.5) 8. Sae auchy s residue heorem. (Tex Book Page No.:5.83) 9. Evaluae 0. Evaluae c 3 7 d, where is d, where is he circle /. ( )( ). Using auchy s inegral formula, evaluae. Evaluae d, where is he circle. (Tex Book Page No.:5.8) sin cos d, where is.. (Tex Book Page No.:5.8) 3 3. Evaluae d, where is (a) (b) 3. (Tex Book Page No.:5.3). If f ( ) ( ) ( )..., find he residue of f( ) a. Tex Book Page No.:5.7 Sri Hariganesh Publicaions (Ph: , ) Page 6

7 Engineering Mahemaics Idenify he ype of singulariies of he following funcion: e f ( ). Tex Book Page No.: Idenify he ype of singulariy of funcion sin. (Tex Book Page No.:5.7) 7. alculae he residue of e f( ) ( ) a is pole. (Tex Book Page No.:5.69) 8. Find he residue of he funcion f( ) 3 ( ) a a simple pole. (Tex Book Page No.:5.67) 9. Find he residue of f( ) ( )( ) a. (Tex Book Page No.:5.68) 0. Find he residue of e a 0. (Tex Book Page No.:5.70) Texbook for Reference: ENGINEERING MATHEMATIS - II Publicaion: Sri Hariganesh Publicaions Auhor:. Ganesan Mobile: , To buy he book visi All he Bes---- Sri Hariganesh Publicaions (Ph: , ) Page 7

It is easier to visualize plotting the curves of cos x and e x separately: > plot({cos(x),exp(x)},x = -5*Pi..Pi,y = );

It is easier to visualize plotting the curves of cos x and e x separately: > plot({cos(x),exp(x)},x = -5*Pi..Pi,y = ); Mah 467 Homework Se : some soluions > wih(deools): wih(plos): Warning, he name changecoords has been redefined Problem :..7 Find he fixed poins, deermine heir sabiliy, for x( ) = cos x e x > plo(cos(x)