Dronacharya Group of Institutions, Gr.

Noida
Department of Applied Sciences (First Year)

Even Semester (2020-2021)

Subject Name & Code: Engineering Mathematics (KAS 203T)

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Unit : 01 , Differential Equations
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1. The order of the differential equation 30 is

2. What are the order and degree of the differential equation =0

d2y
3. The type of differential equation  sin  x  y   sin x is
dx 2
a) Linear, homogenous
b) Non linear, homogenous
c) Linear, non- homogenous
d) Non-linear, non-homogenous
4. The differential equation reprents a family of

5. The number of arbitrary constants in the complete primitive of differential equation

is

6. The particular integral of , is

7. The solution of , satisfying the condition ) is

8. All real solution of the differential equation y  2ay  by  cos x (where a and b are
real constants are ……..if
a) a  1, b  0 b) a  0 & b  1 c) a  1, b  0 d) a  0, 1  1

9. The differential equation whose linearly independent solutions are cos 2 x, sin 2 x and e2 x is

a)  D3  D 2  4 D  4  y  0
 
b)  D3  D 2  4 D  4  y  0
 
c)  D3  D 2  4 D  4  y  0
 
d)  D3  D 2  4 D  4  y  0
 
10. Linear combination of solution of an ordinary differential equal are also solution if the
differential equation is
a) Linear non-homogenous
b) Linear homogenous
c) Non-linear homogenous
d) Non-linear non homogenous
is the general solution of

12. The solution of is

13. The particular integral of is

14. If is a solution of of ,then the second linearly

independent solution of this equation is

d. Constant
15. The number of linearly independent solution of - of
the form ( being a real number) is

1 1
16. The formula sin(ax  b)  sin(ax  b) is applicable only if
2
f (D ) f (a 2 )

a. f (a 2 )  0 b. f (a 2 )  0 c. f (a 2 )  0 d. f (a 2 )  0

17. Suppose y p  x   x cos 2 x is a particular solution of y  y  4 sin 2 x . Then the

constant  equals.
a) 1 b) -2 c) 2 d) 4
18. Particular integral for (4D  4D  3) y  e is
2 2x

1 2x 1 2 x
a. e b. e 2 x c. e d. e2 x
21 21

19. A particular solution of 4 is

d.

d2y dy
20. The solution yx  of the differential equations  4  4 y  0 , satisfying the
dx 2 dx

condition y0  4, 0  8 is

dy
dx

a) 4e
2x b) 16 x  4e 2 x c) 4e
2 x d) 4e
2 x  16 xe 2 x

2 d2y dy
21. The general solution of x  5 x  9 y  0 is
dx 2 dx

c1  c2 x e3x b)  c1  c2 ln x  x3 c)  c1  c2 x  x3 d)  c1  c2 ln x  e x
3
a)

d2y
22. Consider the differential equation  12 x 2  24 x  20 with the condition
dx 2
x  0, y  5 and x  2, y  2. Then the value of y at x  1 is
a) 15 b) 17 c) 18 d) 0
23. Solution of y  y  0, y0  1, y   is
2
1
a) cos x  2 sin x b) cos x  sin x c) cos x  sin x d) cos x  2 sin x
2
 x0  0, y0
dx dy
24. Solution of the simultaneous diff. equation  y,  x; are
dt dt
a) x  0, y  0
b) x  k1, y  k2
c) x  cos t , y  sin t
d) x  cos t , y  any value-
25. The particular integral of is

26. The solution of (here k is a non zero constant),which vanishes when x=0
and which tends to finite limit as x tends to infinity is

d.

27. Particular integral for ( D 2  4) y  cos 2 x is

cos 2 x  sin 2 x  x sin 2 x x sin 2 x

a. b. c. d.
8 2 4 4
28. Particular integral for ( D  1) 3 y  e  x is

1 x 3e  x x 2ex x3
a. e  x b. c. d.
8 8 4 6 ex

29. The general solution of the linear differential equation  D 4  81 y  0 is given by-
 
a)  C1  C2 x  e   C3  C4 x  sin 3x
3x

b)  C1  C2 x  e3x  C3  C4 x  e3x  C5  C6 x  cos3x  C7  C8 x  sin 3x

c) C1e3 x  C2e3 x  C3 cos3x  C4 sin 3x
d) C1e3 x  C2e3 x  e3 x  C3 cos x  C4 sin x 

Qx  is equal to –
1
30.
D 
a) e x  Q  x  dx

b) e 
 x e x Q x  dx

c) e x  Q  x  dx
d) e x  e xQ  x  dx
 The general solution of the differential equation D2  D  1 y  e x is
2
31.

a) y  C1  C2 x  C3  C4 x e x

b) y  C1  C2 x  C3  C4 x e  x  e x
1
4
c) y   C1  C2e x    C3  C4  e  x  e x
1
4
d) None of these
d2y
32. The general solution of the d.e.  y  e x is –
dx 2
a) y  A cos  x  B   e x
1 x
b) y  A cosh  x  B   xe
2
c) y  A cosh  x  B   xe x
d) y  A cos  x  B   xe x
33. The general solution of the differential equations  D2  D  2  y  e x is given by –
1
a) y  C1e x  C2e2 x  xe x
3
b) y  C1e x  C2e2 x
1
c) y  C1e x  C2e2 x  x 2e x
6
1
d) y  xe x   C1  C2 x  e2 x
3
34. The P.I. of the differential equation  D2  4  y  x is –

a) xe
2 x b) x cos 2 x c) x sin 2 x d) x / 4
d2y
35. The solution on the d.e.   2 y  10 2 is –
dx 2
a) y  A cos x  B  10
b) y  A sinx  B   10 2
c) y  Ax  Bx cos x  10 2
d) y  A cosx  B  10

d2y
36. The general solution of the differential equation  a 2 y  sec ax is –
dx 2
a) y  C1 cos ax  C2 sin ax  x sin ax  logcos ax 
 b) y  C1 cos ax  C2 sin ax 
1
x sin ax  logcos ax 

1 
 x sin ax  logcos ax 
1
c) y  C1 cos ax  C2 sin ax 
a a 
1 
 x sin ax  logcos ax  cos ax
1
d) y  C1 cos ax  C2 sin ax 
a a 
   
2
37. The general solution of the differential equation D3  1 y  e x  1 is –

x x  1 1  1
a) y  C1e  e 2  C2 cos 3x  C3 sin 3x   e 2 x  e x  1
 2 2  9
b) y  C1  e
x
2 C2 cos 3x  C2 sin 3x   1 2x
9
e  ex 1

 
x
x 1 1 1
c) y  C1e  e  C2 cos 3x  C3 sin
2
3x   e 2 x  e x
 2 2  3
d) None of these
1
38. The value of cos ax is………….
D2  a2
x x x
a) cos ax b) sin ax c) sin ax d) None of these
2a 2a 2a
d 1
39. If D  , then sin x equals –
dx 2
D  D 1
a)  cos x b) cos x c) cos x  sin x d) sin x
2
d y
40. The general solution of the d.e. 2
 4 y  sin 2 x is given by-
dx
a) y  C1e2 x  C2e2 x  2sin x cos x
1 x
b) y  C1 cos 2 x  C2 sin 2 x   sin 2 x
8 8
x
c) y   C1  C2 cos 2 x  e2 x  cos 2 x
8
d) y  C1 cos2 x  C2  
1
8

d2y
41. The general solution of the differential equation  a  bx  cx 2 given that dy  0
dx 2 dx
when x  0 and y  d when x  0 is –
1 2
 ax  2bx  cx  d 
a) 3 4
12  
 1 
 6ax  2bx  cx  12d 
b) 2 3 4
12  
1 2 1
c) ax  bx3  cx 4  d
12 6
d) None of these
42. The general solution of the differential equation  D3  3D2  2D  y  x 2 is-

x  2 x 2  9 x  21
1
a) C1  C2e x  C3e2 x 
12
b) C1  C2e x  C3e2 x  x  2 x 2  9 x  21
1
12

x  2 x 2  9 x  21
1
c) C1   C2  C3 x  e 
x

12
d) C1  C2e x  C3e2 x  1  2 x 2  9 x  21
12
43. P.I. of the differential equation  D2  2  y  x 2e3 x  e x cos 2 x is –

e x 11x 2  12 x  50   e x  4sin 2 x  cos 2 x 

1 3 1
a)
121 17
1 3x  2 50  1
b) e 11x  12 x    e x  4sin 2 x  cos 2 x 
121  11  17
1 3x  2
e  x  12 x  50   e x sin 2 x  cos 2 x 
1
c)
121   17
d) None of these
44. P.I. of the differential equation  D2  4  y  sin 2 x  e x is –
1 1
a)  x cos x  e x
4 5
1 1
b)  x cos 2 x  e x
4 5
x 1
c)  cos 2 x  e x
2 5
x 1 x
d)  cos 2 x  e
2 3
1
45. ea x is equal to –
  D  D  a 
n

x n e ax
a)
 a 

x ne ax
b)
n !  a 
 x n e ax
c) ,  a   0
 a 

x ne ax
d) ,  a   0
n !  a 
46. The P.I. of  D2  1 y  e x sin x is –

ex
a)  2 cos x  sin x 
5
ex
b) 2 cos x  sin x 
4
ex
c) 2 sin x  cos x 
4
ex
d) 2 sin x  cos x 
5
47. P.I. of the differential equation  D4  2D2  1 y  x 2 cos x is
x3
sin x   9 x 2  x 4  cos x
1
a)
12 48
b)
1 3
12
x cos x 
1
48
 9 x 2  x 4  sin x

x cos x   9 x 2  x 4  sin x
1 3 1
c)
12 16
x cos x   9 x 2  x 4  sin x
1 3 1
d)
6 16
1
If V be any function of x, then xV is equal to
f D 
48.

 f D  
a)  x  f D .V
 f D 
 f ' D  
b)  x  f D .V
 f D  
 f ' D  
c)  x  2 f D   f D .V
 

 f ' D  
d)  x  2 f D   f D .V
 

49. Particular integral for is

50. Solution of the simultaneous differential
dx dy
equations  3 x  8 y,  x  3y is obtained. Then solution
dt dt
is

51. The points x & y lie on ………………..,where x & y are solution for
dx dy
 y,  x .
dt dt

dx dy
52. Solution of the simultaneous differential equations  5x  2 y  t,  2x  y  0 .
dt dt
Given that x  y  0 when t  0 , then

53. Particular solution for the

 d2y dy
54. The differential equation 2
 (3 sin x  cot x)  2 y sin 2 x  sin 2 xe cos x is solved
dx dx
by changing the independent variable into independent variable then we must have

a. z   2 sin x b. z  2 cot x c. z  cos ecx d. z   cos x

55. The differential equation is solved by changing

the Independent variable into independent variable then

a. b. c. d.

d 2v
56. If  Iv  S is the normal form of
dx 2
d2y dy  1 1 6 
 x 1 / 3   2 / 3  4 / 3  2  y  0 obtained by solving change of dependent
dx 2
dx  4 x 6x x 
variable , then the value of I is

a.1 b. 0 c. 6 x 2 d. - 6 x 2

d 2v
57. If  Iv  S is the normal form of -4 then
dx 2
the value of I is

a. 1 b. 0 c. x 2 d. - 6 x 2

58. A part of C.F for y   cot x y   1  cot x y  e x sin x is

a. cot x b. sin x c. e x d. e  x

59. The basis for the equation given that y = cotx is a solution of it, is

d2y
P  x   Q  x  y  R, 1  
dy P Q
60. For a differential equation  0, then one part of
dx 2 dx a a2
complementary function is
a) e
ax b) x
m c) sin x d) cos x
 d2y
Px   Qx  y  R, P  Qx  0, then one part of
dy
61. For differential equation
dx 2 dx
complementary function is
a) e
ax b) x
m c)
1
d) x
2
x
d2y dy
62. Solution of x 2
  3  x   3 y  0 is
dx dx
a) y  c1  x3  3x 2  bx  6   c2e x
b) y   c1  c2 x  e x  3x 2  4 x
1
c) y  c1 x  c2 x   x9
2

x4
d) y  c1  c2 x e x 
1
4
Solution of y  4 xy   4 x 2  2  y  0, given that y  e x is a solution.
2
63.

a) y  x  x  1 e  c1 x  c2 x
2
x ex

2
b) y  e x c1x  c2 
A  1
c) y   c2  x  
x  x
d) None

2 d2y dy
64. The solution of diff. equation x  x  y  0, given that x  1 is an integral.
dx 2 dx x
2 A  1
a) y  e x c1x  c2  c) y   c2  x  
x  x
c
b) y  c1x  2  x 2 d) None
x

d2y dy
65. cos x  sin x  2 y cos 3 x  2 cos 5 x is being solved by changing of independent
dx 2 dx
variable from x into z. Here
a) z  cos x b) z  e
x c) z   sin x d) z  cos x
66. Solving by variation of parameter y  2 y  y  e x log x then the value of wornskion is

a) e
2x b) e
2 x c) 2 d) x
2
67. Complementary function for ( D 2  2) 3 y  0 is………….
 d2y dy
68. The general solution of the equation x 2 2
 x  2 y  0 is ……………
dx dx

69 . Particular integral for the equation is …………..

70. The general solution for ( D  1) 3  sin hx is

71. By changing the independent variable, we get the solution y = ………… of
1
y   y  4 x 2 y  x 4 .
x
72. The reduced normal form of the differential equation
is given by ………..

73.. You are going to solve the given differential

equation ,by changing the independent
variable. The reduced equation with constant coefficients is …………

3/2 2/3
 d3y   d3y 
74. The order and degree of the differential equations  3   3   0 are
 dx   dx 
(A) 3, 3 (B) 3,9 (C) 3, 6 (D) 9, 6

75. The solution of the differential equation  D  1  D  2  y  0 is

2

(A) y  c1  c2 x  c3e2 x (B) y  c1e x  c2 x  c3e2 x

(C) y  c1e2 x  c2 x  c3 (D) Both (A) and (C)

76. The P.I. of  D2  a 2  y  cos ax, where a  0, is

x sin ax x sin ax x cos ax x cos ax

(A) (B)  (C) (D) 
2a 2a 2a 2a

d3y d2y
77. Solution of the differential equation  3  4 y  0 is
dt 3 dt 2

(A) y  c1et   c2  tc3  e2t (B) y  c1e x   c2  xc3  e2 x

(C) y  c1et   c2  tc3  e2t (D) y  c1et   c2  tc3  e2t

78. The P.I. of  D2  1 y  x 2 is

(A)  x 2  2  (B)   x 2  2 

(C)   x 2  2  (D)   x 2  1

 D  2 y  17e2 x is
3
79. The P.I. of

17 3 x 17 2 2 x
(A) xe (B) xe
6 6

17 3 2 x 17 4 2 x
(C) xe (D) xe
6 6

d2y dy
80. The P.I. of differential equation 2
 4  12 y   x  1 e2 x is
dx dx

e2 x  x 2 9 x  e2 x  x 2 9 x 
(A)    (B)   
8  2 7  8  2 8 

e2 x  x3 9 x  e2 x  x 2 9 
(C)    (D) 9   
8  2 8  8  2 8

dx dy
81. The solution of the simultaneous differential equations   y ,   x lies on
dt dt
(A) An ellipse (B) Parabola (C) Hyperbola (D) Circle
Ans. (D)

d2y dy
82. The P.I. of the differential equation 2
 2  y  xe x cos x
dx dx

(A) e x   x cos x  2sin x  (B) e x   x cos x  sin x 

(C) e x  x cos x  2sin x  (D) e x  2 x cos x  sin x 

83. Order of the differential equations is the

(A) highest order derivative involving equation (B) lowest order derivative involving equation
(C) Two derivatives (D) None of these.
84. The degree of the differential equation is the power of highest order derivative involving in
the equation provided the

(A) the differential equation is free from radical signs

(B) the differential equation is free from fractional powers

(C) Both A &B (D) None of these.