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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
MCQ
Choose the correct answer from the given four options.The solution of the differential equation $\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}-\text{y}}+\text{x}^2\text{e}^{-\text{y}}$ is :
  • A
    $\text{y}=\text{e}^{\text{x}-\text{y}}-\text{x}^2\text{e}^{-\text{y}}+\text{c}$
  • ✓
    $\text{e}^{\text{y}}-\text{e}^{\text{x}}=\frac{\text{x}^3}{3}+\text{c}$
  • C
    $\text{e}^{\text{x}}+\text{e}^{\text{y}}=\frac{\text{x}^3}{3}+\text{c}$
  • D
    $\text{e}^{\text{x}}-\text{e}^{\text{y}}=\frac{\text{x}^3}{3}+\text{c}$
✓

Answer

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

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