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DIFFERENTIAL EQUATIONS UNIVERSITY OF CALICUT V Semester CORE COURSE

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DIFFERENTIAL EQUATIONS UNIVERSITY OF CALICUT V Semester CORE COURSE
School of Distance Education
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
B.Sc. MATHEMATICS
V Semester
CORE COURSE
DIFFERENTIAL EQUATIONS
QUESTION BANK
1. The order of the differential equation (
(a) 0
(b) 1
) +
2. The degree of the differential equator ( )
(a) 0
(b) 1
=
(c) 2
=
(c) 2
3. The integral curves of the differential equation
(a)
=
+
(b)
=
+
is
is?
= 1 are ?
(c)
4. Which of the following is a linear differential equation ?
(a)
(c)
+ ( ) =
+ 3
+
Differential Equations
=0
(b) (
(d) (
=
(d)
ℎ
(d)
ℎ
+
) + 3
(d)
=
) + ( ) +
=
+1
=0
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5. Which of the following is a separable differential equation ?
(a)
(c) )
=
+ (
(b)
) =
(d)
=
+ (
) = 0
6. An integrating factor of the differential equation
+2 =4
7. A homogeneous differential equation
=
separable equator using a transformation:
can be converted to a variable
(a)
(b)
(a) =
(c)
=
(b)
8. The differential equation (6
+4
(a)
(c)
)
=
(c)
+ (6
+
(d)
ℎ
9. The general solution of the differential equation 2 (3 +
) = 0 is
(a)
(c))
+
+2
+
+
+
=
=
10. An integrating factor of the differential equation 3(
6
=0 is
(a)
(b)
11. The solution of the differential equation
(a) (0, ∞)
(b) (−∞, 0)
12. Which of the following is an initial value
problem :
(a)
(c)
+
+
= 0,
= 0,
Differential Equations
(0) =
(0) = 0,
(0) = 0
(1) = 1
−
(b)
=
(d)
)
+
(c))
(d)
)
+
(b)
is ?
+
+
ℎ
=
(d)
= 0 is ?
)
+2
+(
+2 +
+ (
+3
+3
+
(d)
+
=
+
=
, (0) = 1 exists in the region
(d) (−∞, ∞ )
(c)) (−∞, 1)
(b)
(d)
+
+
= 0,
= 0,
(0) =
(0) = 0,
(1) = 0
(2) = 4
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13. Which of the following is a boundary value problem :
(a)
(b)
(c)
(d)
+
(0) = 1,
= 0,
+ 5 = 0,
+
+
+
+
(0) = 1,
(0) = 0
(0) = 3
(0) = 0,
= 0,
= 0, (0) =
(0) =
(1) = 2
(0) = 0
14. An integrating factor of the differential equation
(a)
(b)
(c))
(2 −
+ 2( − )
+ 2)
(d)
15. The general form of a first or linear equation is
(a)
+
= where P and Q are functions of
(c)
=
where Q is a functions of
(b)
+
=
where P is a functions of
(d) None of these
16. The general solution of the differential equation
(a)
=
(a)
=
(b)
=
+
=
=
(b)
=
(d)
18. The differential equation Mdx + Ndy = 0 is exact if
(a) M=N
19. If )
(a)
(b)
−
( )=
(c) ) ( ) = ∫
Differential Equations
is afunction of
∫
−
−
=
= cos is
(c)
17. The general solution of the differential equation
(c)
= 0 is
(c)
+
=
(d) y= Sin ( ) +
= 0 is
−
=
only, then an integrating factor of
(b) ) ( ) =
(d) ( ) = ∫
= 0
(d) )
∫
−
+
+
= 0 is
.
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20. If
−
is afunction of y only, then an integrating factor of the differential
equation Mdx+Ndy=0 is
(a) ( ) =
(b) ( ) = ∫
∫
−
(b) ( ) =
−
(d) ( ) = ∫
21. An integrating factor of the differential equation
(a)
∫
(b)
∫
(c)
∫
22. An integrating factor of the differential equator
functions of y alone is
(a)
∫
(b)
23. The initial value problem
(a) a unique solution
(c) no solution
∫
∫
=
+
( ) = ( ) is
(d)
+
=
(c)
∫
−
+
∫(
)
where P and Q are
(d)
, (0) = 0, ≥ 0
∫
(b) infinitely many solutions
(d) two solutions
24. A mathematical model of an object falling in the atmosphere near the surface of earth is
given by
(a)
(c)
= mg-rv
(b)
= mg
(d) None of these
25. The general solution of the differential equation 3(
6
=0 is
(a)
(c)
+3
+3
=
=
26. The domain of the differential equation (1+
(a) (0, ∞)
(c) (−∞, ∞ )
Differential Equations
= mg-rv
+
(b)
(d)
)
) y″ + xy ′ + y =0 is
+
+3
+
(
=
+3
=
+
(b) (−∞, 0)
(d) None of these
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27. If y1( ) and y2( ) are two linearly independent solutions of the linear differential
equation
a0 ( )y ″ + a1( )y ′+ a2 +(x)y = 0 then
(a)
( )
(c)
( )/ ( )
( )+
(b)
( )
( )+
(d)
( )
( )
28. Let y1( ) and y2( ) be two linearly independent solutions of the differential equation a0
( )y ″ + a1( )y ′ + a2 (x)y = 0 then the Wronskian ( , ) is
(a) 1
(b) 0
29. The differential equation
−5
+ 6 = 0 has
(c) 2
(d) − 1
(a) two linearly independent solutions
(b) three linearly independent solutions
(c) four linearly independent solutions
(d) infinite number of linearly independent solution
30. The characteristic equation of the differential equation (D2 – 4D + 4)y = 0 is
( ) ( − 2) = 0
(c) ( − 2) = 0
31. The general solution of the differential equation (
(a) (
(c)
+
)
(b) ( + 2) = 0
(d) ( − 1)( − 2) = 0
− 4 + 4)y=0 is
(b) (
(d)
−
+
)
32. The characteristic roots of the differential equation ( − 8 + 25) = 4
(a)
(b)
(c)
(d)None of these
2 are
33. The characteristic roots of the differential equation ( − 2 ) = 4 + 2 + 3 are
( ) = 0, = −2
(b) = 1, = 3
(c) = 0, = 2
(d) = 1, = −2
Differential Equations
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34. The general solution of the differential equation (
( ) =
[ cos 2 + sin 2 ]
(b) =
cos √2 + sin √2
(c) =
cos √2 +
sin √2
(d)
ℎ
35. A particular solution of the differential equation
( )
2
2 +
(c) cos 2 + sin 2
sin 2
36. A particular solution of the differential equation
( )
(
+ tan )
(c) log (sin + cos )
+ 2 + 3) = 0 is
+4 =
(b)
2
2 is
2 +
Cos 2
(d) log(cos 2 ) + log(sin 2 )
+
(b)
(d)
= tan is
log
(
+ Cos )
37. Let y1(x) and y2(x)be two independent solutions of the differential equation y″ + 4y = 0
Then W(y1, y2) =
(a) 0
(b) 1
(a)
(b)
38. A particular solution of
+5
39. The transformation =
into the following form
(a) (
− 4 + 1) =
(c) (
− 4 + 3) =
ᵼ
ᵼ
+6 =
is
(c) 2
(d) ∞
(c)
(d)e
transform the differential equation
(b) (
−3
+
− 4 + 1) =
(d)(
− 4 + 2) =
ᵼ
=
ᵼ
40. The differential equation
−
−3 =
can be converted into a
differential equation with constant coefficients using the transformation
(a)
=
(a)
=
(b) =
41. The general solution of the differential equator
(c)
=
+
Differential Equations
+
−
(c)
=
z
− 2 = 0 is
(b)
=
(d) y = e
(d)x =
+
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42. The auxiliary equation of the differential equator (
(a) (
(c) (
+ 2) = 0
− 1)(
− 2) = 0
43. A particular integral of the d.e. (
− 2 + 1) =
− 4 + 4) =
(b) (
(d)
(a)
(b) −
(a)
(b)
(c)
sin 3 ]
(b)
44. A particular integral of the differential equation (3
45. The general solution of the differential equation (
( )
[
[
(c)
cos 3 +
Sin +
cos ]
(c)
+
− 2) = 0
is
+1=0
(b) 1, 1
47. The differential equation
+7
48. The general solution of (
− 6 + 9) = 0 is
(a) two independent solutions
(c) four independent solutions
(a) (
+
( )
=
)
− 8 = 0 has
(b) (
(c)
=
Sin √2 +
Cos
50. The general solution of (
( )
(c)
=[
=
cos √3 +
cos √5 +
Differential Equations
Cos √2
− 4 + 13) = 0 is
[
cos 3 +
[cos 3 + sin 3 ]
(d)
cos √5
−2
(c) −1, −1
+
]
=
(d)1, 0
(d) only one independent solution
+
)
(b)
(d)
+ 4 + 7) = 0 is
sin √3 ]
(d)x
(b) three independent solutions
49. The general solution of the differential equation
Sin +
(d)e
− 14) = 13
46. The roots of the auxiliary equation of the differential equation
are
(a) 2, 2
is
(c) (
+
=
+ )
= 0 is
Sin 2 +
= Sin + Cos
(b)
=
(d)
Cos 2
cos √2 +
(d) cos √3 + sin √3
sin √2
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51. The Laplace transform of the unit step function (
(a)
(b)
(a)
(b)
52. The Laplace transform of
is
53. The Laplace transform of cosat is
(a)
(b)
54. If ℒ{ ( )} = F(s), then ℒ {
( )} =
55. If ℒ{ ( )} = F(s), then ℒ { (
)} =
(a) F(s)
(a)
(b) F(s-a)
f(s/a)
(b) F(s/a)
(c) )
=
57. ∫
(a)
=
58. ∫
(a)
( )
Differential Equations
(d) )
(d) None of these
(c)
(d)
(c) F(s +a)
(d) F(s/a)
(c) F(a/s)
(d) F(s)
/
(c)
(b)
(c)
(d)
(b)
(c)
(d) None of these
(b)
( )
(c)
{ ( − )} =
( ⁄ )
(d)
/
(b)
59. If F(s) is the Laplace transform of f(t) then ℒ
(a)
is
(c)
56. The Laplace transform of the delta function is
(a)
)
(d)
( ⁄ )
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60. ℒ
61.
ℒ
(a) sin3t
(a)
(c)
+5
(b)
(c)
=
(b)
+5
(d)
(d) cos 3t
−5
62. If ℒ{ ( )} = F(s) and ℒ{ ( )} =G(s), then ℒ{ ∗ } =
63. ℒ {
64. ℒ
−5
(a) F(s) G(s)
(b) F(s) + G(s)
(c) F(s) – G(s)
(d) F(s)/G(s)
(a)
(b)
(c)
(d)
(b) t3
(c) t2
(d)
2
(c) ( )
(d)
( )
∗ cos } =
=
(a)
2
65. If ℒ{ ( )} = F(s), then ℒ
(a)
( ⁄ )
Differential Equations
{ (
)} =
(b) ( ⁄ )
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66. The Laplace transform of the function whose graph shown below is
k
f(t)=k, t > 1
F(t) = kt
(a)
0
(1 −
)
(c) (1 −
)
67. ℒ {sin ℎ
}=
(a)
68. ℒ {cos ℎ
(a)
69. ℒ {
70. ℒ
}=
(a)
t
!
(a) 4t sin 2t
Differential Equations
)
(d) None of these
}=
2 +4 2
(b) (1 −
(b)
(c)
(d)
(b)
(c)
(d)
(b)
=
(
)!
(b) sin 2t
(c)
!
(c) t sin 2t
(d)
(d) sin 4t
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71. ℒ {
(a)
(c)
(
(
72. . ℒ
(
)
)
(b)
(2 + )
]
(d) None of these
, then ℒ{ e
(a)
(2 + )
(b)
(2 + )
73. . ℒ{ sin 4 } =
sin 4 } =
(c)
(b)
(d) None of these
74. The Laplace transform of the function f(t) =
(b)
[(
)
=
3
(c)
(a)
(
)
(d) None of these
)
−2
(a)
}=
sin
(
(
)
)
1−
[1 −
(
)
]
75. If f(t) | − 1| + | + 1|, then ℒ{ ( )} =
0
0< <1
>1
(b)
(d)
(a)
1−
(b)
(c)
1−
(d)
Differential Equations
(
is
)
[1 −
1−
(
)
]
1−
(1 −
)
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76. The Eigen values of the BVP : y″ + 2y = 0, y (0) = y(π) = 0 are
(a) 1, 4, 9, …………..
(b) 0, 2, 4, 6, 8, ………
(c) 1, 2, 3, …………..
(d) 2, 5, 7 ……….
77. The eignen functions of he BVP: y″ + 7 y = 0, y(0) = y (π) = 0 are
(a) Sin nx, n = 1, 2, 3, ………..
(b) Cos nx, n = 1, 2, 3, ……….
(c) tan nx, n = 1, 2, 3, ………..
78. The eigen functions of the BVP:
(d) Cot nx, n = 1, 2, 3, ……...
y″+λy = 0, y(0) = y(L) = 0 are
(a) yn(x) =
, n = 1, 2, 3, ……..
(c) yn(x) =
, n = 1, 2, 3, ……..
(b) yn(x) =
, n = 1, 2, 3, ……..
(d) yn(x) =
, n = 1, 2, 3, ……..
79. The eigen value of the BVP: y″+λy = 0, y(0) = y(L) = 0 are
(a) λn =
(c) λn =
(
, n = 1,2, 3……..
)
, n = 1,2, 3……..
80. The period of the function sin nx is
(a) 2π
81. The functions sin
(a) [-1, 1]
Differential Equations
(b) 2π/n
and cos
b) [-π, π]
(b) λn =
(d) λn =
(c) n
, n = 1,2, 3……..
, n = 1,2, 3……..
(d)None of these
, m = 1, 2, 3, …… are orthogonal is the interval
(c) [-L, L]
(d)[-∞,∞]
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82. Which of the following function is an odd function.
(a) f(x) = cos x
(b) f(x) = sin x
83. Which of the following is an even function
(a) f(x) = cos x
(b) f(x) = tan x
(c) f(x) = ex
(d) f(x) = x4
(c) f(x) = sin x
(d) f(x) = ex
84. Let f(x) = | |, -2 ≤ x ≤ 2, f(x + 4) = f(x) for all x ∊ ( -∞, ∞). Then the Fourier sine
coefficient bn is given by
(a) 0
(b) 1/π2
(c) 1/n2 π2
(d) 1/n2
85. Let f(x) = | |, -2 ≤ x ≤ 2, f(x + 4) = f(x) for all x ∊ (-∞, ∞). Then the Fourier cosine
coefficient an is given by
(a) 0
(c)
(b) nπ
[ (−1) − 1]
(d) -1
86. Which of the following is a partial differential equation:
(b) uxx +uyy= 0
(a) y″ +λy = 0
(c) y‴+ y″ + y′ + y = 0
87. One dimensional heat equation is
(a) ∝
(c) ∝
=
=
,0 <
,0 <
88. The solution of the PDF:
< , >0
< , >0
(d) yiv + ( ) = ex
(b)
(d)
=
=
,0 <
,0 <
< , >0
= 12 xy2 + 8x3 e2y , ux (x, 0) = 4x, u(0,y) = 3 is
(a) 2x2y3 + x4 e2y + 2x2 – x4 + 3
(c) 2x2y3 - x4 e2y + 2x2 – x4 + 3
< , >0
(b) 2x3y3 + x4 e2y - 2x2 + x4 + 3
(d) 2xy + x4 e2y + 2x3 – x4 + 3
89. The Fourier series of the function f(x) = x, - π ≤ x ≤ π, f(x + 2π) = f(x) is
(a) a constant function
(c) a cosine series in x
Differential Equations
(b) an identity function
(d) a sine series in x
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90. The period of the function sinx + cos x is
(a) 2π
(b) 4π
91. Which of the following functions are periodic
(a) x
(b) sinx
(c) 3π
(d) 0
(c) x2
(d) x4
92. Which of the following function is neither odd or even.
(a) sinx
(b) cos x
(c) sin hx
93. If f(x) and g(x) are even functions, the
(a) f(x) g(x) is even
(b) f(x) g(x) is odd
(c) f(x) + g(x) is even
94. The problem
=
(d) f(x) – g(x) is odd
, u(0,t) = 0, u(L, t) = 0 is
(a) an initial value problem
(c) an initial Boundary Value Problem
95. The problem
=
(c) An Initial Boundary Value Problem
=
u(0, t) = 0
u(L, t) = 0
(a) An initial value problem
(b) An initial boundary value problem
Differential Equations
(b) a boundary value problem
(d) None of these
, u(x,0) = f(x), ut (x, 0) = g(x), 0 ≤ x ≤ L is
(a) An Initial value problem
96. The problem
(d) tan x
(b) a Boundary Value Problem
(c) None of these
u(x,0) = f(x)
ut(x, 0) = g(x), 0 ≤ x ≤ L is
(b) a boundary value problem
(d) None of these
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97. The solution to the heat conduction problem
=
, 0 < x < 50, t > 0
u(x, 0) = 20, 0 < x < 50
u(0, t) = 0, u(50, t) = 0, t > 0 is
∑
(a) u (x, t) =
∑
(b) u (x, t) =
∑
(c) u (x, t) =
∑
(d) u (x, t) =
, , ,…..
/
/
/
, , ,…..
/
98. The one dimensional wave equation is given by
(a)
(c)
=
, 0 < x < L, t > 0
= ∝
, 0 < x < L, t > 0
(b)
(d)
=
=
, 0 < x < L, t > 0
, 0 < x < L, t > 0
99. The solution to the boundary value problem;
=
u(0,t) = 0
, 0≤x≤L
u(L, t) = 0 is given by
(a) u(x, t) = ∑
(b) u(x, t) = ∑ , , ,…..
(c) u(x, t) = ∑
+
+
(d) u(x, t) = ∑
Differential Equations
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School of Distance Education
100. The solution to the problem
=
u(0, t) = 0
(a) u(x, t) = ∑
u(L, t) = 0
, 0≤x≤L
u(x,0) = f(x)
ut(x, 0) = 0 is given by
(b) u(x, t) = ∑
(c) u(x, t) = ∑
(d) u(x, t) = ∑
Differential Equations
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School of Distance Education
ANSWER KEY
1.
c
23.
a
45.
d
3.
a
25.
b
47.
a
2.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
c
c
a
c
a
c
a
c
b
a
c
a
a
d
a
c
a
a
a
a
Differential Equations
24.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
a
c
b
b
a
c
a
b
a
d
a
a
a
b
a
a
b
b
a
a
46.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
b
b
a
a
b
a
a
b
a
a
b
c
a
b
a
a
a
a
a
a
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67.
a
79.
a
91.
b
69.
c
81.
c
93.
a
68.
70.
71.
72.
73.
74.
75.
76.
77.
78.
b
a
b
a
a
a
a
a
a
a
80.
82.
83.
84.
85.
86.
87.
88.
89.
90.
b
b
a
a
92.
94.
95.
96.
c
b
a
a
d
97.
98.
99.
100.
c
c
a
c
a
a
a
a
a
©
Reserved
Differential Equations
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