Question

The solution of $\frac{dy}{dx}+\frac{y\cos x+\sin y+y}{\sin x+x\cos y+x}=0$ is
(where $C$ is an arbitrary constant)

• A. $y\sin x+x\sin y+xy=C$
• B. $x\sin x+y\sin y+xy=C$
• C. $y\cos x+x\cos y+xy=C$
• D. $(x\cos x+y\cos y)xy=C$

Answer 1

The correct option is A $y\sin x+x\sin y+xy=C$

Given :
$\frac{dy}{dx}+\frac{y\cos x+\sin y+y}{\sin x+x\cos y+x}=0\Rightarrow [\sin x+x\cos y+x]dy+[y\cos x+\sin y+y]dx=0\Rightarrow (\sin xdy+y\cos xdx)+(x\cos ydy+\sin ydx)+(xdy+ydx)=0\Rightarrow d\left(y\sin x\right)+d\left(x\sin y\right)+d\left(xy\right)=0$
integrating both sides,
$\Rightarrow \int d\left(y\sin x\right)+\int d\left(x\sin y\right)+\int d\left(xy\right)=0\Rightarrow y\sin x+x\sin y+xy=C$