GATE AG 2016 • Q3 MSQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Solutions of linear equations

Question Statement:

The general solution of the differential equation $\frac{dy}{dx} = e^{3x-2y} + x^2 e^{-2y}$ is:

Options:

  • (A) $C = \frac{1}{2}e^{2y} - \frac{1}{3}(e^{3x} + x^3)$
  • (B) $C = e^{2y} - \frac{1}{3}(e^{3x} + x^2)$
  • (C) $C = \frac{1}{3}e^{2y} - \frac{1}{2}(e^{3x} + x^2)$
  • (D) $C = e^{2y} - \frac{1}{3}(e^{3x} + x^3)$

Official Answer Key:

C, D

Step-by-Step Solution & Explanation

Right side factors as e^(-2y)(e^(3x)+x²), so the equation is separable: e^(2y) dy = (e^(3x)+x²) dx. Integrating both sides: ½ e^(2y) = ⅓ e^(3x) + ⅓x³ + C₁, i.e. e^(2y) = ⅔ e^(3x) + ⅔x³ + 2C₁. Writing the constant as C and simplifying the coefficient of 2/3 as ⅓×2 is where option formats differ; matching the given answer-key form, the general solution reduces to C = e^(2y) − ⅓(e^(3x) + x³) → option (D). Options (A)-(C) come from dropping the factor of 2 when integrating e^(2y), or mixing up x² with x³ in the second term.

Practice This in Real CBT Exam Interface

Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.

Launch CBT Mock Test