Question

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

• A. x = c sin (y/x)
• B. x = c sin (xy)
• C. y = c sin (y/x)
• D. xy = c sin (x/y)

Answer 1

Answer: (A)

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