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, $\frac{dy}{dx}=\frac{y}{x}+\tan \frac{y}{x}$
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
$$\begin{gathered} \Rightarrow \log c+\log \sin v=\log x \\ c\sin v=x \end{gathered}$$
$\Rightarrow x=c\sin \left(\frac{y}{x}\right)$