GATE AG 2008 • Q8 MCQ • 1 Mark

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)

Question Statement:

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

Options:

  • (A) $y = \sqrt{\frac{2}{3}x^3 + 4x}$
  • (B) $y = \sqrt{\frac{2}{3}x^3 + 4x + k}$
  • (C) $y = \sqrt{\frac{2}{3}x^3 - 4x + k}$
  • (D) $y = \frac{2}{3}x^3 + 4x + k$

Official Answer Key:

B

Step-by-Step Solution & Explanation

(A) y = √((2/3)x³ + 4x) (B) y = √((2/3)x³ + 4x + k) (C) y = √((2/3)x³ − 4x + k) (D) y = (2/3)x³ + 4x + k Separate variables: y dy = (x²+2) dx. Integrate: y²/2 = x³/3 + 2x + C → y² = (2/3)x³ + 4x + k → y = √((2/3)x³ + 4x + k).

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