64 m kj izns'k jktf"kZ V.Mu eqDr fo'ofo|ky;] bykgkckn

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28 89 m kj izns’k jktf"kZ V.Mu eqDr fo’ofo|ky;] bykgkckn fo’k; % xf.kr Subject : Mathematics dkslZ “kh’kZd % Differntial equation Course Title : Differntial equation vf/kU;kl (Assignment) 2012-2013 Lukrd dyk dk;ZØe Under Graduate Art Programme Section - A [k.M & d fo’k; dksM% ;w0th0,e0,e0 Subject Code : UGMM dkslZ dksM % ;w0th0,e0,e0-08 Course Code : UGMM-08 vf/kdre vad % 30 Maximum Marks:30 vf/kdre vad % 18 Max. Marks: 18 uksV % nhkZ mRrjh; iz”uA vius iz”uksa ds mRrj 800 ls 1000 “kCnksa esa fy[ksaA lHkh iz”u vfuok;Z gSA Note : Long Answer Questions. Answer should be given in 800 to 1000 words. Answer all questions. 1(a) dy Solve: cos( x + y) = 1 dx 3 dy gy djsa cos( x + y) = 1 dx (b) 2 Solve: ( x + sin y) dx + ( x cos y - 3y ) dy = 0 3 2 gy djsa ( x + sin y) dx + ( x cos y - 3y ) dy = 0 2(a) Solve: + x sin 2y = x cos 2 y dx 3 gy djsa + x sin 2y = x cos 2 y dx (b) Find general and singular solution for (p xy) (xpy) = 2p. 3 dy where p = dx Lkk/kkj.k ,oa fofp= gy Kkr djsaA 2 put ( x 2 = 4 , y = v ) 3 ''' ' 2 3(a) Solve: x y + 2xy - 2y = x log x + 3x 3 gy djsa 3 ''' ' 2 x y + 2xy - 2y = x log x + 3x '' (b) Solve: y + 4 y = 4 tan 2x by method of variation of parameters. 3 gy djsa ¼oSfj;lu vkWQ ikjk ehVjl½ fof/k ls Section - B [k.M & [k vf/kdre vad % 12 Max. Marks: 12 uksV % ykq mRrjh; iz”uA vius iz”uksa ds mRrj 200 ls 300 “kCnksa esa fy[ksaA lHkh iz”u vfuok;Z gSA Note : Short Answer Questions. Answer should be given in 200 to 300 words. Answer all questions. '' 2 4 Solve: y + y = sin x by method of undetermined coefficients. 3 gy djsa '' 5 Solve: y = e x cosh x , 1 ' 1 y (0) = , y (0) 3 8 4 gy djsa 2 2 d y d y 6 Solve: - 3x - 4y = 0 and + x + y = 0 2 2 dt gy djsa 7 Find the equation of oblique trajectory of family of circles 3 2 2 x + y - 2cy = 0 at 60¢ 2 2 o`Rrksa ds lewg + y - 2cy = 0 dt x dk 60¢ ij vkscfyd VªSftDVjh dks Kkr djsaA 3
29 90 m kj izns’k jktf"kZ V.Mu eqDr fo’ofo|ky;] bykgkckn fo’k; % xf.kr Subject : Mathematics dkslZ “kh’kZd % Real Analysis Course Title : Real Analysis vf/kU;kl (Assignment) 2012-2013 Lukrd dyk dk;ZØe Under Graduate Art Programme Section - A [k.M & d fo’k; dksM% ;w0th0,e0,e0 Subject Code : UGMM dkslZ dksM % ;w0th0,e0,e0-09 Course Code : UGMM-09 vf/kdre vad % 30 Maximum Marks:30 vf/kdre vad % 18 Max. Marks: 18 uksV % All qestions ar compulsory Note : Section A is long Answer and Section B is Short Answer Questions. 1(a) Show that Z/ × Z/ is countable 3 (b) Define a bounded set S. Find that S = { Î IR 2 £ x £ 31 } (a) (b) 2(a) x is bounded. What are the least upper bound and greatest lower bound. 3 fn[kk;s fd Z/ × Z/ ,d x.krh; leqPp; gSA ,d ifjc) leqPp; S dks ifjHkkf"kr djsa rFkk crk;sa fd S = { ÎTR 2 £ x £ 31 } x ,d ifjc) leqPp; gS bldk vf/kdre rFkk U;wure c) Kkr djsaA ì Test the convergence of the sequence í în 2 2 ü ý + 2n + 3þ æ 1 ö (b) Test the convergence of the serise å cos ç ÷ 3 è n ø 3 (a) ì 2 ü vuqØe í ý ds vfHklkjh gksus ds fy, ijh{k.k djsaA 2 în + 2n + 3þ (b) æ 1 ö Js.kh å cos ç ÷ ds vfHklkjh dh lR;rk dks Li"V djsaA è n ø 3(a) 2 lim 1- cos2x + (2cos x) x Evaluate: 2 x ® 0 x 3 Kkr djsa ¥ n+ 1 (-1) (b) Show that the series å is conditionally convergent. 3n + 2 3 fn[kk;s fd Js.kh å ¥ n- 1 n-1 (-1) 3n + 2 n+ 1 lizfrcU/k vfHklkjh gSA. Section - B [k.M & [k vf/kdre vad %12 Max. Marks:12 4 æ 1 ö Show that f(x) = sinç ÷ is not uniformly continuous on (0,1). è x ø 3 æ 1 ö fn[kk;sa fd f(x) = sin ç ÷ ,d lekur% (0,1) ij lrr ugha gSA è x ø 5 State and prove cauchy mean value theorem. 3 dks’kh ek/;eku izes; dks fy[kdj fl) djsaA 6 3 2 4 3 2x 2 x 2 x5 Show that : e x cos x = 1 + x - - - + ......... 3! 4! 5! 3 3 2 4 3 2x 2 x 2 x5 fn[kk;sa fd e x cos x = 1+ x - - - + ......... 3! 4! 5! 7 For a bounded function f on [a,b] Show that 3 b x1 x2 ò f ( x) dx = ò f ( x) dx + ò f ( x) dx + ò a a x1 b x2 f ( x) dx
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Page 3 and 4: 3 5. Derive the wave equation for a
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87 6. efJeÅeeLeea meneÙelee mesJe
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153 OTHER INFORMATION : Volume 2 :
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64 m kj izns'k jktf"kZ V.Mu eqDr fo'ofo|ky;] bykgkckn

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