JEE (Main) Mathematics - Differential Equations MCQs

46. 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
47. 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
48. 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) \( \frac{dy}{dx} + 3y = 0 \)
49. 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) \( y = Ce^{-x} + x^2 \)
50. 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 \( \frac{dy}{dx} = g(x)h(y) \)
51. 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
52. 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) \( y = Cx \)
53. 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) \( x^3 \)
54. 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) \( (x + y)dx + (x - y)dy = 0 \)
55. 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) \( y = Ce^{-4x} \)
56. 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
57. 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) \( y = 2e^{-x} + 1 \)
58. 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
59. 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) \( y = Ce^x + x \)
60. 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) \( \frac{dy}{dx} = f(x)g(y) \)

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