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 x 2 1 e 1 dx dy dx d y + + = (By method of variation of parameters) (iv) x 2 e (D +3D+2)y=e Q.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 x dx dy x dx d y x 2 12 log 3 2 2 2 + − = Q. 3.Solve the following Linear Differential Equations: (i) 36y 3x 4x 1 dx dy 3(3x 2) dx d y (3x 2) 2 2 2 2 + + + − = + + (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 2 2 2 + + + = x + dx dy x dx d y x Q. 4 The equations of motion of a particle are given by: wy 0 dt dx + = , wx 0. dt dy − = Find the path of the particle. Q.5 Solve: 2 3x 6y t dt dx − − = ; 3y e . dt dx dt dy t + − = Q.6 Solve: (i) 2 2 2 z a x y z dz y dy x dx − + + = = (ii) xy z dz y dy x dx 2 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) = 0 2) The solution of the auxiliary eq. is called--------- a) C.F. b) P.I. c) Complete solution d) None 3) If the C.F. involves 5 distinct constants then order of corresponding D.E.is--- a) 3 b) 4 c) 5 d) >5 4) If yc=c1+c2 x+c3 x2 then A.E. is ----- a) (D33D2+3D1)=0 b) D3=0 c) D2=0 d) D=0 5)The C.F. of the D.E. (D2 y+D y )= 1 1+ex is---- a) c1+c2 ex b) c1+c2 ex c) c1+c2 ex d ) (c1 x+c2)ex 6)The C.F. of the D.E. d 2 y dx2 +y=cosec x is---- a) c1+c2 cos( x) b) c1+c2 ex c c1cos (x )+c2 sin( x) d ) c1e x+c2 ex 7) The P.I. Of the D.E. D y = x is----- a) x d x b) x c) x2 d x d) none 8) The P.I. Of the D.E. (D3+3D) y=cosh(x ) is---- a)2 cosh( x) b) 1 2 cosh( x) c) 1 4 sinh (x) d) 1 4 sinh (x ) 9)The P.I. Of the D.E. (D2) y=x+1 is----- a) x+1 b) x1 2 c) x+1 2 d) x1 10)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)none 11)The P.I. Of the D.E. (D2+4) y=x Is -------- a) x 4 b) x 8 c) x 2 d) x 4 12)By using substitution x=e z the 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)=0 13)By MVP the solution of the D.E. (D2+1) y= 1 1+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 x d t + y=sin(t ) , d y d t +4x=cos (t ) the C.F. for x is------ a) c1e2 t+c2 e3 t b) c1e2t+c2 et c) c1e2 t+c2 e2 t d) c1e2 t+c2 et 15)The solution of the D.E. dx x (2y4z4) = dy y ( z 42x4) = dz z (x4y4) by using the multipliers x3 , y3 , z3 are------------ a) (x+y+z )=c1 b) (x2+ y2+z2)=c1 c) (x3+y3+z3)=c1 d) (x4+y4+z 4)=c1

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 x
2
1 e
1
dx
dy
dx
d y
+
+ = (By method of variation of parameters)
(iv)
x 2 e (D +3D+2)y=e
Q.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 x
dx
dy
x
dx
d y
x 2 12 log 3
2
2
2
+ − =
Q. 3.Solve the following Linear Differential Equations:
(i) 36y 3x 4x 1
dx
dy
3(3x 2)
dx
d y
(3x 2) 2
2
2
2
+ + + − = + +
(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 2
2
2
+ + + = x +
dx
dy
x
dx
d y
x
Q. 4 The equations of motion of a particle are given by:
wy 0
dt
dx
+ = , wx 0.
dt
dy
− =
Find the path of the particle.
Q.5 Solve:
2 3x 6y t
dt
dx
− − = ; 3y e .
dt
dx
dt
dy t
+ − =
Q.6 Solve: (i)
2 2 2 z a x y z
dz
y
dy
x
dx
− + +
= = (ii)
xy z
dz
y
dy
x
dx
2 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) = 0
2) The solution of the auxiliary eq. is called---------
a) C.F. b) P.I. c) Complete solution d) None
3) If the C.F. involves 5 distinct constants then order of corresponding D.E.is---
a) 3 b) 4 c) 5 d) >5
4) If yc=c1+c2 x+c3 x2
then A.E. is -----
a) (D33D2+3D1)=0 b) D3=0 c) D2=0 d) D=0
5)The C.F. of the D.E. (D2 y+D y )= 1
1+ex is----
a) c1+c2 ex
b) c1+c2 ex
c) c1+c2 ex
d ) (c1 x+c2)ex
6)The C.F. of the D.E.
d 2 y
dx2 +y=cosec x
is----
a) c1+c2 cos( x) b) c1+c2 ex
c c1cos (x )+c2 sin( x) d ) c1e x+c2 ex
7) The P.I. Of the D.E. D y = x is-----
a) x d x b) x c) x2 d x d) none
8) The P.I. Of the D.E. (D3+3D) y=cosh(x ) is----
a)2 cosh( x) b)
1
2
cosh( x) c)
1
4
sinh (x) d)
1
4
sinh (x )
9)The P.I. Of the D.E. (D2) y=x+1 is-----
a) x+1 b)
x1
2 c)
x+1
2 d) x1
10)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)none
11)The P.I. Of the D.E. (D2+4) y=x Is --------
a)
x
4 b)
x
8 c)
x
2 d)
x
4
12)By using substitution x=e z
the 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)=0
13)By MVP the solution of the D.E. (D2+1) y= 1
1+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 x
d t
+ y=sin(t ) ,
d y
d t
+4x=cos (t ) the C.F. for x is------
a) c1e2 t+c2 e3 t
b) c1e2t+c2 et
c) c1e2 t+c2 e2 t
d) c1e2 t+c2 et
15)The solution of the D.E.
dx
x (2y4z4)
= dy
y ( z 42x4)
= dz
z (x4y4) by using the multipliers
x3 , y3 , z3
are------------
a) (x+y+z )=c1 b)
(x2+ y2+z2)=c1 c)
(x3+y3+z3)=c1 d)
(x4+y4+z 4)=c1