Q98 / 150 · MHT CET 2026 (15 April Shift 1) · Mathematics

The solution of the differential equation $\dfrac{dy}{dx} = e^{x-y} + x^2 e^{-y}$ is...

  • A
    $y = e^x + c$
  • B
    $e^y = e^x + x^3 + c$
  • Correct answer
    C
    $e^y = e^x + \dfrac{x^3}{3} + c$
  • D
    $e^y = e^x + 2x + c$

Solution

The given differential equation is $\dfrac{dy}{dx} = e^{x-y} + x^2 e^{-y}$

This can be rewritten by factoring out $e^{-y}$ as $\dfrac{dy}{dx} = e^{-y}(e^x + x^2)$

Separating the variables, we get $e^y dy = (e^x + x^2) dx$

Integrating both sides, we get $\int e^y dy = \int (e^x + x^2) dx$

$e^y = e^x + \dfrac{x^3}{3} + c$

Answer: $e^y = e^x + \dfrac{x^3}{3} + c$

Hence, the correct answer is option: 'C'.

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