Question

The solution of the differential equation $\cos x\sin ydx+\sin x\cos ydy=0$ is

• A. $\sin x+\sin y=c$
• B. $\sin x\sin y=c$
• C. $\frac{\sin x}{\sin y}=c$
• D. $\cos x\cos y=c$

Answer 1

The correct option is B $\sin x\sin y=c$

$\text{Given}\cos x\sin ydx+\sin x\cos ydy=0$
$\Rightarrow \cos x\sin ydx=-\sin x\cos ydy$
$\Rightarrow \tan ydx=-\tan xdy$
$\Rightarrow \cot xdx=-\cot ydy$
Integrating on bothsides,
$\Rightarrow \int \cot xdx=-\int \cot ydy$
$\Rightarrow \ln |\sin x|=-\ln |\sin y|+k$
$\Rightarrow \ln |\sin x|+\ln |\sin y|=k$
$\Rightarrow \ln |\sin x\sin y|=k$
$\Rightarrow |\sin x\sin y|=e^k$
$\Rightarrow \sin x\sin y=\pm e^k=c$
Hence option (b) is correct.