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GATE 2025 CE (CE2) – Question 36

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 2 marks · Multiple choice

Pick the CORRECT solution for the following differential equation
$\frac{dy}{dx} = e^{x-y}$

  1. $y = \ln(e^x + Constant)$
  2. $\ln(y) = x + Constant$
  3. $\ln(y) = \ln(e^x) + Constant$
  4. $y = x + Constant$

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Correct answer: (A) $y = \ln(e^x + Constant)$

Explanation

Write $\frac{dy}{dx} = e^xe^{-y}$ and separate the variables: $e^y\,dy = e^x\,dx$. Integrating, $e^y = e^x + C$, so $y = \ln(e^x + C)$.