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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 solution of the differential equation $\cos\text{x}\ \sin\text{y}\ \text{dx}+\sin\text{x}\ \cos\text{y}\ \text{dy}=0$ is:
  1. $\frac{\sin\text{x}}{\sin\text{y}}=\text{C}$
  2. $\sin\text{x}\ \sin\text{y}=\text{C}$
  3. $\sin\text{x}+\sin\text{y}=\text{C}$
  4. $\cos\text{x}\ \cos\text{y}=\text{C}$
✓

Answer

  1. $\sin\text{x}\ \sin\text{y}=\text{C}$

Solution:

Given differential equation is

$\cos\text{x}\ \sin\text{y}\ \text{dx}+\sin\text{x}\ \cos\text{y}\ \text{dy}=0$

$\Rightarrow\cos\text{x}\ \sin\text{y}\ \text{dx}=-\sin\text{x}\ \cos\text{y}\ \text{dy}=0$

$\Rightarrow\frac{\cos\text{x}}{\sin\text{x}}\text{dx}=-\frac{\cos\text{y}}{\sin\text{y}}\text{dy}$

$\Rightarrow\cot\text{x}\ {\text{dx}}=-\cot\text{y}\ {\text{dy}}$

On integrating both sides, we get

$\log\ \sin\text{x}=-\log\ \sin\text{y}+\log\ \text{C}$

$\Rightarrow\log\ \sin\text{x}\ \sin\text{y}=\log\ \text{C}$

Carrying the exponent on both sides, we get

$\Rightarrow\sin\text{x}\ \sin\text{y}=\text{C}$

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