BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination. June, 2017 ELECTIVE COURSE : MATHEMATICS MTE-08 : DIFFERENTIAL EQUATIONS

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1 No. of Printed' Pges : 8 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Exmintion n June, 017 ELECTIVE COURSE : MATHEMATICS MTE-08 : DIFFERENTIAL EQUATIONS MTE-08 I Time : hours Mximum Mrks : 50 (Weightge : 70%) Note : Question no. 1 is compulsory. Answer ny four questions out of the remining questions no. to 7. Use of clcultors is not llowed.. 1. Stte whether the following sttements re True or Flse. Justify your nswer with the help of short proof or counter-exmple. 5x=10 () (b) dv = (x + y) is liner first order differentil eqution. dy = y with y(0) = 1 hs unique solution. (c) If X(x, y, z) is vector such tht X. curl X = 0, then [IX. curl 1.1X = 0, where [t(x, y, z) is n rbitrry function of x, y, z.,k A (d) x I" = e c t is two dimensionl wve eqution. MTE-08 1 P.T.O.

2 (e) The norml form of the differentil eqution dy x dy + xy = exi is d V x/ + V = 1, where V = ye. () Solve the differentil eqution z z + IA = sin (x - y). x y x y (b) (c) Find the solution of the differentil eqution (xy4 + xy) + (xy - x) dy = 0. Using the method of vritions of prmeters, find the generl solution of the differentil eqution for 0 < x < -TE d y + y = cot x.. () Solve : dy py + 4px = y, where p = (b) Obtin the solution of the following simultneous differentil equtions : dy dz x - yz y - xz z - xy (c) Find the complete integrl of the eqution u XZ + u z + u u = 0 using Jcobi's method. MTE-08 4

3 x 4. () Use the method of reduction of order to solve the following eqution : 5 (D + 1) y = cosec x (b) A certin popultion is known to be growing t rte given by the logistic d eqution = x (b - x), where nd b re dt positive constnts. Show tht the mximum rte of growth will occur when the popultion is equl to hlf the equilibrium size, tht is, when the popultion is ( b () Using the method of undetermined coefficients write the form of prticulr integrl of the eqution (b) (c) d d x Solve the differentil eqution = x + cos x + e - x. (1 + y) - (tn-1 y - x) dy = O. 4 Verify tht the eqution ( - yz) + (yz - xz) dy - (x - xy + y) dz = 0 is integrble nd find its integrl () A steel bll weighing 9. kg is suspended from spring due to which the spring is stretched m from its nturl length. The bll is strted in motion with no initil velocity by displcing it through 0.5 m bove the equilibrium position. Assuming no ir resistnce, find n expression for the position of the bll t ny time t. 4 MTE-08 P.T.O.

4 (b) Using the trnsformtion x = 1, reduce the differentil eqution y = c4p _ Xp, (p = dy to differentil eqution of the Clirut's form. Hence, solve it. (c) Show tht the prtil differentil equtions p q _ 1 = 0 nd ( p q) x _ pz = 0 re comptible. 7. () Clssify whether the following eqution is elliptic, prbolic or hyperbolic. Justify your nswer. y z z z y Z x z + x = + xe EY x y y x x y y (b) Use the method of seprtion of vribles to show tht the solution of the Lplce,4,,, eqution =-!.; + =!," = 0 together with the x- y conditions (1)(0, y) = 4)(, y) = 0, (1)(x, 0) = 0, nicx 4)(x, b) = sin is 4(x, y) = n = 1 ( nicx nxy sin sinh sinh nib 8 MTE-08 4

5 74.tt chltist0-1 (4t.fr.rfts.) HAiti trft k.kelW 1 ckt0t) 4 1 Icl Tht.t.-08 rdwr chvi R-IT4s : W" 1P-1----f *: 50 (yff : 70%) 7iF 1,wi gf l.? TR7 T g 7 # 17--gf yr( Jr / 4(-5de7' Nw-177- cr,4 4i` 7-1Tr& 4V7 1. (gcl.tt-4tip, tzfr ucrea- wt Nc-jqWul 1 ki61111 A. T'1.irk ttr A7I 5x=10 dy () = (x + y) 7114 E trtfff 4-4 t I dy = y,116 y(0) = 1, I X(x, y, z) Lkch TfiT4T 161. X. curl X = 0, "-q. x, y, z * lb-w4 u(x, y, z) *. curl Pi = 0. (v) 4) 1 (1) f4int (1M lc of t I x c t MTE-08 5 P.T.O.

6 1 (t) TT*-e ti d y chul x d y + x y = e x/ d V + V = 1 t, 111. ( ) v = ye- x/ t.141chkut x z " To- 1-rir Alf-4R I T-4T-o- ch(ui z z = sin (x - y) yx (xy4 + xy) + (xy - x) dy = o.11-rr Al-P- A-R -9-m f-4-4tur 1-4RT ir Siel) 1 1 cb() o < x < T-4-*7. kii-nchkui dy + y = cot.x 0,-Ii4ch c rrc- 41.N7 : p y + 4px = y, \lw p = PiHroRgi TrEr-q 1-4w-.11-ficMui1 dy dz y x yz y xz z xy -9-Rr*=rr-A-R (T) 1* WET Mei 4 1 -K 4)01 u 1 xz + u z + u u 0 k :11c1 711c1 Ar-A-R 1 MTE

7 4. 5. () ocr)(ul f -RT -srzul-rf (.1c c11(ul WA7 : ti+fl (D + 1) y = cosec x A-R-Its,e4 t t -F4- ;fil fze1tff chkui dt = x (b - x), z0 1 b T- T t, qri 1-R1 I f-qtgr-47 6 TRT-*--dIT It ff4 *-d1 74 zfriki(sql 41ffq 170 W). crielk *-41. t, 1 b prh, -74 **Rs:ell t4t t ) f-avi-ft7 d y feit -Wr 4 dy =x+cosx+ e -x 14.4)7 kplicho f-ai7 I (u ) >kui (1+ y) - (trr l y - x) dy 0 ttr77 (TO trr-a-r -FT (xz - yz) + (yz - xz) dy - (x - xy + y) dz 0 t ;no trra 4 6. () ict) ITTT JT t, v,ct) ITT r1 ift. *1 114 t t I 4 ic (1d-T-FIPTTT) :11. i 1,1R--TTgrd- 1TR -rf Trrei I -I H q1 I 4-i M pt 6 lrrtv -0., f fi11 of 04,1c1-) TIM *11-4R I 4 MTE-08 7 P.T.O.

8 (W) 1 to-ildkur x = t y = x4p _ xp, (p = dy dx 11 *'tr-a7 qat i 4-flc ul w-u-ild i I 1m -44 (Tr) Iqur-4-R1-- choi p q _ = 0 A{ (p ci) x _ pz = 0 Th7 t I 7. () ki414,kut teiltzr, liko eict, TT--r Trw-d-RTT -iff-t-u*a-r k Y z xy z + x 8 z = y oz + x oz y y x x y y x frt gild fqur-47 fll-ncmul + (I) o srf-d-4x11 xe y 4(0, y) = 4(, y) = 0, 4)(x, 0) = 0, 4(x, b) = sin nnx TfIlT E ( sin n/ex sink IITEY 4)(x, y) = t I 8 sink n7b n = 1 MTE ,000