JEE Main Mathematics: Differential Equations MCQs with Answers

61. The order of the differential equation \( \frac{d^2y}{dx^2} + 3y = 0 \) is:

a) 1 b) 2 c) 3 d) 0

Answer: b) 2

62. The degree of the differential equation \( \left( \frac{dy}{dx} \right)^2 + y^2 = x \) is:

a) 1 b) 2 c) 0 d) Undefined

Answer: b) 2

63. Which of the following is a first-order linear differential equation?

a) \( \frac{dy}{dx} + y^2 = 0 \) b) \( \frac{dy}{dx} + 3y = 0 \) c) \( \frac{d^2y}{dx^2} + y = 0 \) d) \( \frac{dy}{dx} + y = x^2 \)

Answer: b) \( \frac{dy}{dx} + 3y = 0 \)

64. The general solution of the differential equation \( \frac{dy}{dx} + y = x^2 \) is:

a) \( y = Ce^{-x} + x^2 \) b) \( y = Ce^{x} + x^2 \) c) \( y = Ce^{-x} - x^2 \) d) \( y = Ce^{x} - x^2 \)

Answer: a) \( y = Ce^{-x} + x^2 \)

65. Which of the following is the correct method to solve a separable differential equation?

a) Multiply both sides by the integrating factor b) Express the equation as \( \frac{dy}{dx} = g(x)h(y) \) c) Integrate both sides directly d) None of the above

Answer: b) Express the equation as \( \frac{dy}{dx} = g(x)h(y) \)

66. The equation \( \frac{dy}{dx} = \frac{y}{x} \) is an example of:

a) Separable equation b) Linear equation c) Homogeneous equation d) Exact equation

Answer: a) Separable equation

67. The solution to the differential equation \( \frac{dy}{dx} = \frac{y}{x} \) is:

a) \( y = Ce^{x} \) b) \( y = Cx \) c) \( y = Cx^2 \) d) \( y = C \ln(x) \)

Answer: b) \( y = Cx \)

68. For the equation \( \frac{dy}{dx} + \frac{3}{x}y = x^2 \), the integrating factor is:

a) \( x^3 \) b) \( e^{3x} \) c) \( x^{-3} \) d) \( e^{3x} \)

Answer: a) \( x^3 \)

69. Which of the following differential equations is exact?

a) \( (x + y)dx + (x - y)dy = 0 \) b) \( y dx + x dy = 0 \) c) \( (x^2 + y)dx + (x + y^2)dy = 0 \) d) None of the above

Answer: a) \( (x + y)dx + (x - y)dy = 0 \)

70. The general solution of the differential equation \( \frac{dy}{dx} + 4y = 0 \) is:

a) \( y = Ce^{-4x} \) b) \( y = Ce^{4x} \) c) \( y = Cx^4 \) d) \( y = Cx \)

Answer: a) \( y = Ce^{-4x} \)

71. The method of integrating factor is used to solve which type of differential equation?

a) Separable equation b) Exact equation c) First-order linear differential equation d) Homogeneous equation

Answer: c) First-order linear differential equation

72. The particular solution of the differential equation \( \frac{dy}{dx} + y = 1 \) with the initial condition \( y(0) = 2 \) is:

a) \( y = 2e^{-x} + 1 \) b) \( y = 1 + Ce^{-x} \) c) \( y = 2e^{x} + 1 \) d) \( y = e^{-x} + 1 \)

Answer: a) \( y = 2e^{-x} + 1 \)

73. The equation \( \frac{dy}{dx} + \frac{3}{x}y = 0 \) is:

a) Homogeneous linear differential equation b) Non-homogeneous linear differential equation c) Separable differential equation d) Exact differential equation

Answer: a) Homogeneous linear differential equation

74. The general solution of the differential equation \( \frac{dy}{dx} = y + x \) is:

a) \( y = Ce^x + x \) b) \( y = Ce^{-x} + x \) c) \( y = Ce^{x} - x \) d) \( y = Ce^{x} + x^2 \)

Answer: a) \( y = Ce^x + x \)

75. The method of separation of variables can be applied to a differential equation of the form:

a) \( \frac{dy}{dx} = y + x \) b) \( \frac{dy}{dx} = f(x)g(y) \) c) \( \frac{dy}{dx} = y^2 + x^2 \) d) None of the above

Answer: b) \( \frac{dy}{dx} = f(x)g(y) \)

JEE (Main) Mathematics MCQs (Questions 61-75)

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