Question

The solution of the differential equation $\frac{dy}{dx}=\tan \left(\frac{y}{x}\right)+\frac{y}{x}$ is:

• A. $\cos \left(\frac{y}{x}\right)=cx$
• B. $\sin \left(\frac{y}{x}\right)=cx$
• C. $\cos \left(\frac{y}{x}\right)=cy$
• D. $\sin \left(\frac{y}{x}\right)=cy$

Answer 1

Answer: (B)

$\sin \left(\frac{y}{x}\right)=cx$

$\frac{dy}{dx}=\tan \left(\frac{y}{x}\right)+\left(\frac{y}{x}\right)$ ..... $\left(i\right)$
Take, $\frac{y}{x}=v$
$⟹y=vx$
$⟹\frac{dy}{dx}=v+x\frac{dv}{dx}$
$\therefore$ The given equation $\left(i\right)$ becomes
$v+x\frac{dv}{dx}=\tan v+v$
$⟹\frac{1}{\tan v}dv=\frac{1}{x}dx$
$⟹\int \cot vdv=\int \frac{1}{x}dx$
$⟹\log |\sin v|=\log x+\log c=\log |xc|$
$⟹\sin v=xc$
$\therefore \sin \left(\frac{y}{x}\right)=xc$