DEARMEN OF MAEMAICS BI, MESRA, RANCI MA Advad Egg. Mathatis Sssio: S/ 7 MODULE I. Cosidr th two futios f utorial Sht No. -- ad g o th itrval [,] a Show that thir Wroskia W f, g vaishs idtially. b Show that f ad g ar ot liarly idpdt.. Show that th solutios, ad of y y y ar liarly idpdt ad h or othrwis solv th quatio.. Show that is th gral solutio of y y o ay itrval ot otaiig, ad fid th partiular solutio for whih y ad y 5 4. Show that th Wroskia of th futios, log is o zro. Ca ths futios b idpdt solutios of ordiary dfrtial quatios; so dtri this quatio. Solv th followig dfrtial quatios: 5. 5 4 si 6. si 7. si 4 8. D D os 9. D 4 si. a s a. 4 si d y dy. oslog silog d d. d y dy 5 d d 4. Vry that solutio. y is o solutio of y y 4, ad fid sod solutio y ad th gral 5. Solv si, giv that ot is o solutio.
6. Vry that o of th solutio of th quatio is, ad fid aothr solutio valid i. Solv th followig dfrtial quatios by trasforatio or hag of idpdt variabl. 7. y ot y si os os 8. y 4 8 si os 9. si ot y si si. y ta y os MODULE II Solv th followig dfrtial quatios by rduig to Noral For:.. y. y ta y s Solv th followig dfrtial quatios by th variatio of paratrs: 4. ot ot si 5. 6. y y y y ' s ta, / 7. y ''' y s 8. y '''- 6 y '' y '- 6 9. Fid th sigular poits of th followig dfrtial quatios ad lassy th: a b y y y y Fid th sris solutio of th followig dfrtial quatios:. 4 about =. 9 y 4 about = about =. y 5 4. 4 y y about = 4. y y y with y, y 4 about =
MODULE III 5. Driv th solutio of Bssl s quatio of ordr : ad prss th i trs of Bssl s futios 6. rov that a J si b J os d d 7. rov that a [ J ] J b [ J ] J d d 8. rov that a J J J b J J J 9. rov that a J J b J { J J 4. rov that J si os 5 4. rov that 48 4 J J J 4 d 4. Show that J J J J d 4. Show that J d J J 44. If α ad β ar th roots of, th show that J J J d, J, 45. rov that for itgral, J Cos Si d 46. a J b d 47. rov th followig: a b a b 48. Eprss 4 6 i a sris of Lgdr s polyoials. d 49. rov that! d
5. Show: 5. rov that, /! /!, for odd for v 5. rov that d,, 5. rov that d 54. rov that t d t t 55. Driv th solutio of yprgotri quatio: [ a b ] ab, a, b, ostats, i trs iit ovrgt sris about ad prss th i trs of yprgotri futios as : y F a, b, a. b ;! a a b b!....,,,... y F a, b, ;,, 4... ad th gral solutio solutios as y F a, b, ; F a, b, ;, ot a itgr. 56. rov that a F,, ; b l F /,, / ; 57. Driv th solutio of followig dfrtial quatio i trs of yprgotri futios: y 6 y rov th followig rlatios for Chbyshv polyoial os os : 58. a b 4 t rov th followig rlatios usig th gratig futio for as t, t. t t 59. b
6. a ' b 6. rov th orthogoality of Chbyshv polyoials as : d ; rov th followig rlatios for th rit polyoial d d, : 6. a 4 b 8 rov th followig rlatios usig th gratig futio for as t t t! 6. a b 64. rov th orthogoality of rit polyoials as! d *******************************************************************************
MA ADVANCED ENGINEERING MAEMAICS 4 Modul I Liar Dfrtial Equatios of ighr Ordr with ostat offiits. Dfrtial Equatios with Variabl Coffiits: Rduibl to Equatios with ostat Coffiits:- Cauhy Eulr Dfrtial Equatio. Lgdr Liar Dfrtial Equatio. Solutio of Dfrtial Equatio by trasforatio or, hag of Idpdt Variabl. Modul II Rdutio of Dfrtial Equatio to Noral For by rduig th ordr of th Dfrtial Equatio. Solutio by th thod of Variatio of aratrs. Sris Solutio of dfrtial Equatios. Modul III Bssl s dfrtial Equatio ad its Gral solutio. roprtis of Bssl s Futios. Lgdr s Dfrtial Equatio ad Lgdr olyoials. roprtis of Lgdr olyoials. Itrodutio to yprgotri Equatio. Itrodutio to rit Dfrtial Equatios. Itrodutio to Chbyshv olyoials. Modul IV Eig Valus ad Eig Futios. Stur Liouvill robls. riodi Futios. Dirihlt s Coditios o Fourir Sris. Eulr s Forula for Fourir Coffiits. alf rag Sris. Fourir Sris of Futios with Arbitrary priod. Modul V Futio of Copl Variabls. Liit, Cotiuity, Dfrtiability of a Copl Variabl. Cauhy Ria Dfrtial Equatios i Cartsia ad olar Fors. Aalyti Futio. Cauhy-Goursat tho, Cauhy s Itgral hor. Cauhy s Itgral Forula. Morra s thor. Modul VI Liouvill s thor. h fudatal thor of algbra. aylor Sris, Laurt Sris. Rsidus, ols, Rsidu hor. Rsidu at ols. Cotour Itgrals of rigootri to ad Algbrai Equatios to. Coforal Mappig ad Biliar rasforatio. Modul VII artial Dfrtial Equatio, Lagrag s Mthod, oogous ad No-hoogous ighr Ordr Equatios with Costat Coffiits. O-disioal at Equatio. O disioal Wav Equatio t Books:. Sios G. F., Dfrtial Equatios with Appliatios ad istorial Nots. M, d d.,.. R. V. Churhill ad J. W. Brow, Copl Variabls ad Appliatios, 8 th d., 9, MGraw ill.. Dis G. Zill, Warr S. Wright, Advad Egirig Mathatis,4 th d. 4. E. Kryszig, Advad Egirig Mathatis, Jho Wily ad Sos, 8 th d., 999 Rfr Books:. Edwards ad y, Dfrtial Equatios ad Boudary Valu robls, arso Edu., rd d.. Shply L. Ross, Dfrtial Equatios Wily Idia vt. Ltd., rd d.. Birkhoff ad Rota, Ordiary Dfrtial Equatios, Wily Idia vt. Ltd., 4 th d. 4. Zill, Dfrtial Equatios, hoso Larig, 5 th d., 4 5. A.D. Wush, Copl Variabls with Appliatios, arso Eduatio I., rd d. 6. M.J. Ablowitz ad A.S.Fokas, Copl Variabls Itrodutio ad Appliatios, Cabridg ts, d d.