Question

The solution of the differential equation $\left(x\sin \frac{y}{x}\right)dy=(y\sin \frac{y}{x}-x)dx$ is:
(where $c$ is integration constant)

• A. $\cos \frac{y}{x}=\ln |x|+c$
• B. $\sin \frac{y}{x}=\ln |x|+c$
• C. $\cos \frac{y}{x}=2\ln |x|+c$
• D. $\sin \frac{y}{x}=3\ln |x|+c$

Answer 1

The correct option is A $\cos \frac{y}{x}=\ln |x|+c$

$\left(x\sin \frac{y}{x}\right)dy=(y\sin \frac{y}{x}-x)dx$
Putting, $y=vx$
$\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$
$\Rightarrow dy=vdx+xdv$
The equation transforms to:
$\Rightarrow \sin v(vdx+xdv)=(v\sin v-1)dx$
$\Rightarrow \sin vdv+\frac{dx}{x}=0$
Integrating both sides, we get
$-\cos v+\ln |x|=c_1$
$\Rightarrow \cos \frac{y}{x}=\ln |x|+c$
where $(-c_1=c)$