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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 general solution of $\frac{\text{dy}}{\text{dx}}=2\text{x}\ \text{e}^{\text{x}^{2}-\text{y}}$ is:

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

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

  3. $\text{e}^{\text{y}}=\text{e}^{\text{x}^{2}}+\text{C}$

  4. $\text{e}^{\text{x}^{2}+\text{y}}=\text{C}$

✓

Answer

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

Solution:

We have $\frac{\text{dy}}{\text{dx}}=2\text{x}\ \text{e}^{\text{x}^{2}-\text{y}}$

$\Rightarrow\text{e}^{\text{y}}=\frac{\text{dy}}{\text{dx}}=2\text{x}\ \text{e}^{\text{x}^{2}}$

$\Rightarrow\int\text{e}^{\text{y}}\text{dy}=2\int\text{x}\text{e}^{\text{x}^{2}}\text{dx}$

Put $\text{x}^2=\text{t}$ in R.H.S. integral we get

$2\text{x}\text{dx}=\text{dt}$

$\Rightarrow\int\text{e}^{\text{y}}\text{dy}=\int\text{e}^{\text{t}}\text{dt}$

$\Rightarrow\text{e}^{\text{y}}=\text{e}^{\text{t}}+\text{C}$

$\Rightarrow\text{e}^{\text{y}}=\text{e}^{\text{x}^2}+\text{C}$

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