Q104 / 180 · COMEDK 2015 · Mathematics

If $\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x}$, then $\sin \left(\frac{y}{x}\right)=$

  • A
    $c x^{2}$
  • Correct answer
    B
    $c x$
  • C
    $c x^{3}$
  • D
    $\log x$

Solution

We have, $\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x} \quad \text{...(i)}$ Given, differential equation is in homogeneous form. $\therefore$ put $y=v x$ in Eq. (i), we get $v+x \frac{d v}{d x}=v+\tan v \Rightarrow \frac{1}{\tan v} d v=\frac{d x}{x}$ Taking integration on both sides, we get $\log (\sin v)=\log x+\log c$ $$ \Rightarrow \log \frac{\sin v}{x}=\log c \Rightarrow \sin \left(\frac{y}{x}\right)=x c $$

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

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