Question

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

• A. $c{x}^2$
• B. $cx$
• C. $c{x}^3$
• D. $\log x$

Answer 1

Answer: (B)

We have, $\frac{dy}{dx}=\frac{y+x\tan \frac{y}{x}}{x}$ ...(i)
Given differential equation is in homogeneous form
$\therefore$ Put $y=vx$ in (i), we get
$v+x\frac{dv}{dx}=v+\tan v$
$\Rightarrow \frac{1}{\tan v}dv=\frac{dx}{x}$
Taking integration on both sides, we get
$\log (\sin v)=\log x+\log c$
$\Rightarrow \log \frac{\sin v}{x}=\log c$
Solution of the differential equation is $\sin \left(\frac{y}{x}\right)=xc$