Question

The solution of differential equation $\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-2 x\right) d x$ is

• A. $\sin \left(\frac{y}{x}\right)=2 \ln x+c$
• B. $\cos \left(\frac{y}{x}\right)=\ell n x+c$
• C. $\sin \left(\frac{y}{x}\right)=\ell n x+c$
• D. $\cos \left(\frac{y}{x}\right)=2 \ell n x+c$

Answer 1

Answer: (D)

$\sin \left(\frac{y}{x}\right) \frac{d y}{d x}=\frac{y}{x} \sin \frac{y}{x}-2$
Let $y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d y}{d x}$
$ \sin v\left(v+x \frac{d v}{d x}\right)=v \sin v-2$
$x \sin v \frac{d v}{d x}=-2$
$ \sin v d v=-2 \frac{d x}{x}$
$-\cos v=-2 \ell n x+c \Rightarrow \cos v=2 \ell n x+c $
$ \Rightarrow \cos \frac{y}{x}=\ell n x^2+c$