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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 Page 1 School of Distance Education 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 Page 2 School of Distance Education 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 . Page 3 School of Distance Education 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 Page 4 School of Distance Education 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 Page 5 School of Distance Education 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 = + Page 6 School of Distance Education 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 Page 7 School of Distance Education 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) ( ⁄ ) Page 8 School of Distance Education 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) ( ⁄ ) Page 9 School of Distance Education 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 Page 10 School of Distance Education 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 − ) Page 11 School of Distance Education 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)[-∞,∞] Page 12 School of Distance Education 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 Page 13 School of Distance Education 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 Page 14 School of Distance Education 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 Page 15 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 Page 16 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 Page 17 School of Distance Education 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 Page 18