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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 option.
The general solution of $\text{e}^{\text{x}}\cos\text{ydx}-\text{e}^\text{x}\sin\text{ydy}=0$ is:
  1. $\text{e}^{\text{x}}\cos\text{y}=\text{k}$
  2. $\text{e}^{\text{x}}\sin\text{y}=\text{k}$
  3. $\text{e}^{\text{x}}=\text{k}\cos\text{y}$
  4. $\text{e}^{\text{x}}=\text{k}\sin\text{y}$
✓

Answer

  1. $\text{e}^{\text{x}}\cos\text{y}=\text{k}$

Solution:

Given is, $\text{e}^{\text{x}}\cos\text{ydx}-\text{e}^\text{x}\sin\text{ydy}=0$

$\Rightarrow\text{e}^{\text{x}}\cos\text{ydx}=\text{e}^\text{x}\sin\text{ydy}$

$\Rightarrow\frac{\text{d}\text{y}}{\text{d}\text{x}}=\tan\text{ydy}$

$\Rightarrow\text{dx}=\tan\text{ydy}$

On integrating both sides, we get

$\text{x}=\log\sec\text{y}+\text{C}$

$\Rightarrow\text{x}-\text{C}=\log\sec\text{y}$

$\Rightarrow\sec\text{y}=\text{e}^{\text{x}-\text{c}}$

$\Rightarrow\frac{1}{\cos\text{y}}=\frac{\text{e}^\text{x}}{\text{e}^\text{c}}$

$\Rightarrow\text{e}^{\text{x}}\cos\text{y}=\text{k}$ $[\text{where},\text{K}=\text{e}^\text{c}]$

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