Choose the correct answer from the given four option. The solutio… - Vidyadip

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:

$\frac{\sin\text{x}}{\sin\text{y}}=\text{C}$
$\sin\text{x}\ \sin\text{y}=\text{C}$
$\sin\text{x}+\sin\text{y}=\text{C}$
$\cos\text{x}\ \cos\text{y}=\text{C}$

✓

Answer

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

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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