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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS1 Mark
Question
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:

  1. $\text{y}=\text{e}^{\text{x}-\text{y}}-\text{x}^2\text{e}^{-\text{y}}+\text{c}$

  2. $\text{e}^{\text{y}}-\text{e}^{\text{x}}=\frac{\text{x}^3}{3}+\text{c}$

  3. $\text{e}^{\text{x}}+\text{e}^{\text{y}}=\frac{\text{x}^3}{3}+\text{c}$

  4. $\text{e}^{\text{x}}-\text{e}^{\text{y}}=\frac{\text{x}^3}{3}+\text{c}$

✓

Answer

  1. $\text{e}^{\text{y}}-\text{e}^{\text{x}}=\frac{\text{x}^3}{3}+\text{c}$

Solution:

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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