Q. 1. Solve the following Linear Differential Equations:(i) (D 1)y sin(2x 3) 3+ = + (ii) (D 1)y sin(e ) e cos(e ) 2 −x −x −x− = −(iii) 2 x21 e1dxdydxd y++ = (By method of variation of parameters)(iv)x 2 e (D +3D+2)y=eQ.2 Solve the following Linear Differential Equations:(i) (D 1)y sin x sin 2x 2+ = (ii) (D 5D 6)y e sec x(1 2tan x) 2 2x 2+ + = +−(iii) (D 4D 8)y 2+ + 12 e sin x sin 3x −2x=(iv) y x xdxdyxdxd yx 2 12 log 3222+ − =Q. 3.Solve the following Linear Differential Equations:(i) 36y 3x 4x 1dxdy3(3x 2)dxd y(3x 2) 2222+ + + − = + +(ii) (D 4) y x sinh 2x 2+ = (iii) (D 4D 4)y 3xe sin 2x 2 2x− + =(iv) (D 1)y secx 2+ = (By method of variation of parameters)(v) ( 1) ( 1) 2 3 222+ + + = x +dxdyxdxd yxQ. 4 The equations of motion of a particle are given by:wy 0dtdx+ = , wx 0.dtdy− =Find the path of the particle.Q.5 Solve:2 3x 6y tdtdx− − = ; 3y e .dtdxdtdy t+ − =Q.6 Solve: (i)2 2 2 z a x y zdzydyxdx− + += = (ii)xy zdzydyxdx2 4 2 2−=−=Multiple Choice Questions:1) The auxiliary eq. of the D.E. f(D) y = x is------a) f(D) = x b)f(D) y = 0 c) f(D) x = y d)f(D) = 02) The solution of the auxiliary eq. is called---------a) C.F. b) P.I. c) Complete solution d) None3) If the C.F. involves 5 distinct constants then order of corresponding D.E.is---a) 3 b) 4 c) 5 d) >54) If yc=c1+c2 x+c3 x2then A.E. is -----a) (D33D2+3D1)=0 b) D3=0 c) D2=0 d) D=05)The C.F. of the D.E. (D2 y+D y )= 11+ex is----a) c1+c2 exb) c1+c2 exc) c1+c2 exd ) (c1 x+c2)ex6)The C.F. of the D.E.d 2 ydx2 +y=cosec xis----a) c1+c2 cos( x) b) c1+c2 exc c1cos (x )+c2 sin( x) d ) c1e x+c2 ex7) The P.I. Of the D.E. D y = x is-----a) x d x b) x c) x2 d x d) none8) The P.I. Of the D.E. (D3+3D) y=cosh(x ) is----a)2 cosh( x) b)12cosh( x) c)14sinh (x) d)14sinh (x )9)The P.I. Of the D.E. (D2) y=x+1 is-----a) x+1 b)x12 c)x+12 d) x110)The P.I. Of the D.E. (D2+4) y=x cos(2x) Is --------a) x cos2x b) x sin 2x c) x2 sin2 x d)none11)The P.I. Of the D.E. (D2+4) y=x Is --------a)x4 b)x8 c)x2 d)x412)By using substitution x=e zthe A.E. of D.E. (x2 D23xD+5) y=x2 sin( log(x )) is---a) (D24D5)=0 b) (D24D+5)=0 c) (D25)=0 d) (D24D)=013)By MVP the solution of the D.E. (D2+1) y= 11+sin( x) are C.F.= c1cos (x )+c2 sin( x)then value of v is-----a) 1+sin (x) b) log(1+sin(x )) c) 1+cos (x ) d) log(1+cos( x))14)For the D.E.d xd t+ y=sin(t ) ,d yd t+4x=cos (t ) the C.F. for x is------a) c1e2 t+c2 e3 tb) c1e2t+c2 etc) c1e2 t+c2 e2 td) c1e2 t+c2 et15)The solution of the D.E.dxx (2y4z4)= dyy ( z 42x4)= dzz (x4y4) by using the multipliersx3 , y3 , z3are------------a) (x+y+z )=c1 b)(x2+ y2+z2)=c1 c)(x3+y3+z3)=c1 d)(x4+y4+z 4)=c1

Grade 12th pass

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