Question

The solution of $\frac{dy}{dx}=\frac{y}{x}+\tan \frac{y}{x}$ is:

• A. $x=c\sin \frac{y}{x}$
• B. $x=c\sin \left(xy\right)$
• C. $y=c\sin \frac{y}{x}$
• D. $xy=c\sin \frac{x}{y}$

Answer 1

Answer: (A)

$x=c\sin \frac{y}{x}$

$\frac{dy}{dx}=\frac{y}{x}+\tan \frac{y}{x}$
Put $y=vx\Rightarrow \frac{dy}{dx}=x\frac{dv}{dx}+v$
$\therefore x\frac{dv}{dx}+v=v+\tan v$
$\Rightarrow \cot vdv=\frac{1}{x}dx$
On integrating both sides, we get
$\log c+\log \sin v=\log x$
$\Rightarrow c\sin v=x$
$\Rightarrow x=c\sin \left(\frac{y}{x}\right)$