If $\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x}$, then $\sin \frac{y}{x}$ is equal to

• A

  $c x^2$
• Correct answer
  B

  $c x$
• C

  $c x^3$
• D

  $c x^4$

Solution

Given that
This is a homogeneous differential equation. Put $y=v x$ and $\frac{d y}{d x}=v+x \frac{d v}{d x}$ From Eq. (i), $v+x \frac{d v}{d x}=\frac{v x+x \tan \left(\frac{v x}{x}\right)}{x}$ $\Rightarrow \quad x \frac{d v}{d x}=v+\tan v-v$ $\Rightarrow \quad \cot v d v=\frac{d x}{x}$ On integrating both sides, we get $\begin{aligned} \Rightarrow & & \log \sin v & =\log x+\log c \\ \Rightarrow & & \sin v & =x c \\ & & \sin \frac{y}{x} & =x c \end{aligned}$

Hence, the correct answer is option: 'B'.