The GATE Grind

GATE 2024 BT – Question 30

Engineering Mathematics · Differential Equations: First order ODEs, higher order ODEs with constant coefficients, Cauchy and Euler equations, Laplace transforms · 1 mark · Multiple choice

The solution of the differential equation $\frac{dy}{dx} = y + e^{-x}$ that satisfies $y(0) = -\frac12$ is ________.

  1. $-\frac12 e^{-x/2}$
  2. $-\frac12 e^{x}$
  3. $-\frac12 e^{-x}$
  4. $-\frac12 e^{x/2}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) $-\frac12 e^{-x}$

Explanation

The integrating factor is $e^{-x}$: $\frac{d}{dx}(ye^{-x}) = e^{-2x}$, so $y = -\frac12 e^{-x} + Ce^{x}$. $y(0) = -\frac12$ gives $C = 0$.