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Model Paper 10 — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsModel Paper 101 Mark
MCQ
The solution of the differential equation $\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}$ is
  • A
    $e ^{ x }+ e ^{ y }=\frac{x^3}{3}+ c$
  • B
    $e ^{ x }- e ^{ y }=\frac{x^3}{3}+ c$
  • C
    $y=e^{x-y}-x^2 e^{-y}+c$
  • ✓
    $e ^{ y }- e ^{ x }=\frac{x^3}{3}+C$
✓

Answer

Correct option: D.
$e ^{ y }- e ^{ x }=\frac{x^3}{3}+C$
We have, $\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}$
$\Rightarrow e^{y} dy=\left(e^{x}+x^2\right) dx$
$\Rightarrow \int e^{y} d y=\int\left(e^x+x^2\right) d x$
$\Rightarrow e^{y}=e^{x}+\frac{x^3}{3}+c$
$\Rightarrow e^{y}-e^{x}=\frac{x^3}{3}+c$

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