School of Distance Education

Differential Equations Page 1

UNIVERSITY OF CALICUTSCHOOL OF DISTANCE EDUCATION

B.Sc. MATHEMATICS

V Semester

CORE COURSE

DIFFERENTIAL EQUATIONS

QUESTION BANK

1. The order of the differential equation ( ) + = is(a) 0 (b) 1 (c) 2 (d) ℎ2. The degree of the differential equator ( ) = is?(a) 0 (b) 1 (c) 2 (d) ℎ3. The integral curves of the differential equation = 1 are ?(a) = + (b) = + (c) = + (d) = + 14. Which of the following is a linear differential equation ?(a) + ( ) = (b) ( ) + 3 =(c) + 3 + = 0 (d) ( ) + ( ) + = 0

School of Distance Education

Differential Equations Page 2

5. Which of the following is a separable differential equation ?(a) = (b) =(c) ) + ( ) = (d) + ( ) = 06. An integrating factor of the differential equation + 2 = 4 is ?(a) (b) (c) (d) ℎ7. A homogeneous differential equation = can be converted to a variableseparable equator using a transformation:(a) = (b) = (c) = (d) =8. The differential equation (6 + 4 ) + (6 + + ) = 0 is ?(a) (b)(c) (d) ℎ9. The general solution of the differential equation 2 (3 + − ) + ( + 3 +) = 0 is(a) + + 2 + = (b) + + 2 + =(c)) + + = (d) + + 2 + =10. An integrating factor of the differential equation 3( + ) + ( + 3 +6 =0 is(a) (b) (c)) (d)11. The solution of the differential equation = , (0) = 1 exists in the region(a) (0, ∞) (b) (−∞, 0) (c)) (−∞, 1) (d) (−∞,∞ )12. Which of the following is an initial valueproblem :(a) + = 0, (0) = (0) = 0 (b) + = 0, (0) = (1) = 0(c) + = 0, (0) = 0, (1) = 1 (d) + = 0, (0) = 0, (2) = 4

School of Distance Education

Differential Equations Page 3

13. Which of the following is a boundary value problem :(a) + = 0, (0) = 1, (0) = 0(b) + 5 = 0, (0) = 1, (0) = 3(c) + + = 0, (0) = 0, (1) = 2(d) + + = 0, (0) = (0) = (0) = 014. An integrating factor of the differential equation (2 − + 2) + 2( − ) = 0 is(a) (b) (c)) (d)15. The general form of a first or linear equation is(a) + = where P and Q are functions of(b) + = where P is a functions of(c) = where Q is a functions of(d) None of these16. The general solution of the differential equation = cos is(a) = (b) = (c) = (d) y= Sin ( ) +17. The general solution of the differential equation + = 0 is(a) = + (b) =(c) = (d) = −18. The differential equation Mdx + Ndy = 0 is exact if(a) M=N (b) = (c) = (d) ) = 019. If ) − is afunction of only, then an integrating factor of + = 0 is(a) ( ) = ∫ − (b) ) ( ) = ∫ +(c) ) ( ) = ∫ − (d) ( ) = ∫ − .

School of Distance Education

Differential Equations Page 4

20. If − is afunction of y only, then an integrating factor of the differentialequation Mdx+Ndy=0 is(a) ( ) = ∫ − (b) ( ) = ∫ +(b) ( ) = ∫ − (d) ( ) = ∫ −21. An integrating factor of the differential equation + ( ) = ( ) is(a) ∫ (b) ∫ (c) ∫ (d) ∫( )22. An integrating factor of the differential equator + = where P and Q arefunctions of y alone is(a) ∫ (b) ∫ (c) ∫ (d) ∫23. The initial value problem = , (0) = 0, ≥ 0(a) a unique solution (b) infinitely many solutions(c) no solution (d) two solutions24. A mathematical model of an object falling in the atmosphere near the surface of earth isgiven by(a) = mg-rv (b) = mg-rv(c) = mg (d) None of these25. The general solution of the differential equation 3( + ) + ( + 3 +6 =0 is(a) + 3 = (b) + 3 =(c) + 3 = (d) + =26. The domain of the differential equation (1+ ) y″+ xy ′+ y =0 is(a) (0, ∞) (b) (−∞, 0)(c) (−∞,∞ ) (d) None of these

School of Distance Education

Differential Equations Page 5

27. If y1( ) and y2( ) are two linearly independent solutions of the linear differentialequationa0 ( )y ″+ a1( )y ′+ a2 +(x)y = 0 then(a) ( ) ( )(b) ( ) + ( )(c) ( )/ ( )(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 (c) 2 (d) − 129. The differential equation − 5 + 6 = 0 has(a) two linearly independent solutions(b) three linearly independent solutions(c) four linearly independent solutions(d) infinite number of linearly independent solution30. The characteristic equation of the differential equation (D2 – 4D + 4)y = 0 is( ) ( − 2) = 0 (b) ( + 2) = 0(c) ( − 2) = 0 (d) ( − 1)( − 2) = 031. The general solution of the differential equation ( − 4 + 4)y=0 is(a) ( + ) (b) ( − )(c) (d) +

32. The characteristic roots of the differential equation ( − 8 + 25) = 4 2 are(a) (b)(c) (d)None of these33. The characteristic roots of the differential equation ( − 2 ) = 4 + 2 + 3 are( ) = 0, = −2 (b) = 1, = 3(c) = 0, = 2 (d) = 1, = −2

School of Distance Education

Differential Equations Page 6

34. The general solution of the differential equation ( + 2 + 3) = 0 is( ) = [ cos 2 + sin 2 ](b) = cos √2 + sin√2(c) = cos √2 + sin√2(d) ℎ35. A particular solution of the differential equation + 4 = 2 is( ) 2 2 + sin 2 (b) 2 2 + Cos 2(c) cos 2 + sin 2 (d) log(cos 2 ) + log(sin 2 )36. A particular solution of the differential equation + = tan is( ) ( + tan ) (b) ( + Cos )(c) log (sin + cos ) (d) log37. Let y1(x) and y2(x)be two independent solutions of the differential equation y″ + 4y = 0Then W(y1, y2) =(a) 0 (b) 1 (c) 2 (d)∞

38. A particular solution of + 5 + 6 = is(a) (b) (c) (d)e39. The transformation = transform the differential equation − 3 + =into the following form(a) ( − 4 + 1) = ᵼ (b) ( − 4 + 1) = ᵼ(c) ( − 4 + 3) = ᵼ (d)( − 4 + 2) = ᵼ40. The differential equation − − 3 = can be converted into adifferential equation with constant coefficients using the transformation(a) = (b) = (c) = z (d)x =41. The general solution of the differential equator − − 2 = 0 is(a) = + (b) = +(c) = + (d) y = e

School of Distance Education

Differential Equations Page 7

42. The auxiliary equation of the differential equator ( − 4 + 4) = is(a) ( + 2) = 0 (b) ( − 2) = 0(c) ( − 1)( − 2) = 0 (d) + 1 = 043. A particular integral of the d.e. ( − 2 + 1) = is(a) (b)− (c) (d)e44. A particular integral of the differential equation (3 + − 14) = 13(a) (b) (c) (d)x45. The general solution of the differential equation ( − 4 + 13) = 0 is( ) [ cos 3 + sin 3 ] (b) [ cos 3 + ](c) [ Sin + cos ] (d) [cos 3 + sin 3 ]46. The roots of the auxiliary equation of the differential equation − 2 + =are(a) 2, 2 (b) 1, 1 (c)−1,−1 (d)1, 047. The differential equation + 7 − 8 = 0 has(a) two independent solutions (b) three independent solutions(c) four independent solutions (d) only one independent solution48. The general solution of ( − 6 + 9) = 0 is(a) ( + ) (b) ( + ) (c) ( + ) (d)49. The general solution of the differential equation + = 0 is( ) = Sin + Cos (b) = Sin 2 + Cos 2(c) = Sin√2 + Cos√2 (d) = Sin + Cos50. The general solution of ( + 4 + 7) = 0 is( ) = [ cos √3 + sin √3 ] (b) = cos √2 + sin√2(c) = cos √5 + cos√5 (d) cos √3 + sin√3

School of Distance Education

Differential Equations Page 8

51. The Laplace transform of the unit step function ( ) is(a) (b) (c) ) (d) )52. The Laplace transform of is(a) (b) (c) (d) None of these53. The Laplace transform of cosat is(a) (b) (c) (d)54. If ℒ{ ( )} = F(s), then ℒ{ ( )} =(a) F(s) (b) F(s-a) (c) F(s +a) (d) F(s/a)55. If ℒ{ ( )} = F(s), then ℒ{ ( )} =(a) f(s/a) (b) F(s/a) (c) F(a/s) (d) F(s)56. The Laplace transform of the delta function is(a) (b) (c) / (d) /57. ∫ =(a) (b) (c) (d)58. ∫ =(a) (b) (c) (d) None of these59. If F(s) is the Laplace transform of f(t) then ℒ { ( − )} =(a) ( ) (b) ( ) (c) ( ⁄ ) (d) ( ⁄ )

School of Distance Education

Differential Equations Page 9

60. ℒ(a) sin3t (b) (c) (d) cos 3t

61. ℒ =(a) + 5 (b) − 5(c) + 5 (d) − 562. If ℒ{ ( )} = F(s) and ℒ{ ( )} =G(s), then ℒ{ ∗ } =

(a) F(s) G(s) (b) F(s) + G(s) (c) F(s) – G(s) (d) F(s)/G(s)63. ℒ{ ∗ cos } =

(a) (b) (c) (d)64. ℒ =

(a) 2 (b) t3 (c) t2 (d) 265. If ℒ{ ( )} = F(s), then ℒ { ( )} =

(a) ( ⁄ ) (b) ( ⁄ ) (c) ( ) (d) ( )

School of Distance Education

Differential Equations Page 10

66. The Laplace transform of the function whose graph shown below isk f(t)=k, t > 1

F(t) = kt

0 t(a) (1 − ) (b) (1 − )(c) (1 − ) (d) None of these67. ℒ{sin ℎ } =(a) (b) (c) (d)68. ℒ{cos ℎ } =(a) (b) (c) (d)69. ℒ{ } =(a) ! (b) ( )! (c) ! (d)70. ℒ 2+4 2 =

(a) 4t sin 2t (b) sin 2t (c) t sin 2t (d) sin 4t

School of Distance Education

Differential Equations Page 11

71. ℒ{ sin } =(a) ( )( ) (b) ( )[( ) ](c) ( ) (d) None of these

72. . ℒ −2 3 =(a) (2 + ) (b) (2 + )(c) (2 + ) (d) None of these

73. . ℒ{ sin 4 } = , then ℒ{ e sin 4 } =(a) (b)(c) (d) None of these

74. The Laplace transform of the function f(t) = 0 < < 10 > 1 is(a) ( ) 1 − ( ) (b) ( ) 1 − ( )(b) ( ) [1 − ] (d) [1 − ]

75. If f(t) | − 1| + | + 1|, then ℒ{ ( )} =(a) 1 − (b) 1 −(c) 1 − (d) (1 − )

School of Distance Education

Differential Equations Page 12

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, ……….. (d) Cot nx, n = 1, 2, 3, ……...78. The eigen functions of the BVP: y″+λy = 0, y(0) = y(L) = 0 are(a) yn(x) = , n = 1, 2, 3, ……..

(b) yn(x) = , n = 1, 2, 3, ……..(c) 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 = , n = 1,2, 3…….. (b) λn = , n = 1,2, 3……..(c) λn = ( ) , n = 1,2, 3…….. (d) λn = , n = 1,2, 3……..

80. The period of the function sin nx is(a) 2π (b) 2π/n (c) n (d)None of these81. The functions sin and cos , m = 1, 2, 3, …… are orthogonal is the interval(a) [-1, 1] b) [-π, π] (c) [-L, L] (d)[-∞,∞]

School of Distance Education

Differential Equations Page 13

82. Which of the following function is an odd function.(a) f(x) = cos x (b) f(x) = sin x (c) f(x) = ex (d) f(x) = x483. Which of the following is an even function(a) f(x) = cos x (b) f(x) = tan x (c) f(x) = sin x (d) f(x) = ex84. Let f(x) = | |, -2 ≤ x ≤ 2, f(x + 4) = f(x) for all x ∊ ( -∞, ∞). Then the Fourier sinecoefficient bn is given by(a) 0 (b) 1/π2 (c) 1/n2 π2 (d) 1/n285. Let f(x) = | |, -2 ≤ x ≤ 2, f(x + 4) = f(x) for all x ∊ (-∞, ∞). Then the Fourier cosinecoefficient an is given by(a) 0 (b) nπ(c) [ (−1) − 1] (d) -186. Which of the following is a partial differential equation:(a) y″ +λy = 0 (b) uxx +uyy= 0(c) y‴+ y″+ y′ + y = 0 (d) yiv + ( ) = ex87. One dimensional heat equation is(a) ∝ = , 0 < < , > 0 (b) = , 0 < < , > 0(c) ∝ = , 0 < < , > 0 (d) = , 0 < < , > 088. The solution of the PDF: = 12 xy2 + 8x3 e2y , ux (x, 0) = 4x, u(0,y) = 3 is(a) 2x2y3 + x4 e2y + 2x2 – x4 + 3 (b) 2x3y3 + x4 e2y - 2x2 + x4 + 3(c) 2x2y3 - x4 e2y + 2x2 – x4 + 3 (d) 2xy + x4 e2y + 2x3 – x4 + 389. The Fourier series of the function f(x) = x, - π ≤ x ≤ π, f(x + 2π) = f(x) is(a) a constant function (b) an identity function(c) a cosine series in x (d) a sine series in x

School of Distance Education

Differential Equations Page 14

90. The period of the function sinx + cos x is(a) 2π (b) 4π (c) 3π (d) 091. Which of the following functions are periodic(a) x (b) sinx (c) x2 (d) x492. Which of the following function is neither odd or even.(a) sinx (b) cos x (c) sin hx (d) tan x93. 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 (d) f(x) – g(x) is odd94. The problem = , u(0,t) = 0, u(L, t) = 0 is(a) an initial value problem (b) a boundary value problem(c) an initial Boundary Value Problem (d) None of these95. The problem = , u(x,0) = f(x), ut (x, 0) = g(x), 0 ≤ x ≤ L is(a) An Initial value problem (b) a Boundary Value Problem(c) An Initial Boundary Value Problem (c) None of these96. The problem =u(0, t) = 0 u(x,0) = f(x)u(L, t) = 0 ut(x, 0) = g(x), 0 ≤ x ≤ L is(a) An initial value problem (b) a boundary value problem(b) An initial boundary value problem (d) None of these

School of Distance Education

Differential Equations Page 15

97. The solution to the heat conduction problem= , 0 < x < 50, t > 0u(x, 0) = 20, 0 < x < 50u(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) = , 0 < x < L, t > 0 (b) = , 0 < x < L, t > 0(c) = ∝ , 0 < x < L, t > 0 (d) = , 0 < x < L, t > 099. The solution to the boundary value problem;= , 0 ≤ x ≤ Lu(0,t) = 0u(L, t) = 0 is given by

(a) u(x, t) =∑ +(b) u(x, t) =∑ +, , ,…..(c) u(x, t) =∑(d) u(x, t) =∑

School of Distance Education

Differential Equations Page 16

100. The solution to the problem= , 0 ≤ x ≤ Lu(0, t) = 0 u(x,0) = f(x)u(L, t) = 0 ut(x, 0) = 0 is given by(a) u(x, t) =∑(b) u(x, t) =∑(c) u(x, t) =∑(d) u(x, t) =∑

School of Distance Education

Differential Equations Page 17

ANSWER KEY1. c2. c3. a4. c5. a6. c7. a8. c9. a10. c11. b12. a13. c14. a15. a16. d17. a18. c19. a20. a21. a22. a

23. a24. a25. b26. c27. b28. b29. a30. c31. a32. b33. a34. d35. a36. a37. a38. b39. a40. a41. b42. b43. a44. a

45. d46. b47. a48. b49. a50. a51. b52. a53. a54. b55. a56. a57. b58. c59. a60. b61. a62. a63. a64. a65. a66. a

School of Distance Education

Differential Equations Page 18

67. a68. b69. c70. a71. b72. a73. a74. a75. a76. a77. a78. a

79. a80. b81. c82. b83. a84. a85. c86. b87. a88. a89. d90. a

91. b92. c93. a94. c95. a96. c97. a98. a99. a100. a

©Reserved