Question

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

• A. $\sin \frac{x}{y}=cx$
• B. $\sin \frac{y}{x}=cx$
• C. $\sin \frac{x}{y}=cy$
• D. $\sin \frac{y}{x}=cy$

Answer 1

Answer: (B)

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

$\Rightarrow \frac{dy}{dx}=\frac{y}{x}+\tan \left(\frac{y}{x}\right)$

Let $y=vx$

$\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$

$\therefore v+x\frac{dv}{dx}v+\tan \left(v\right)$

$\Rightarrow \frac{dv}{\tan \left(v\right)}=\frac{dx}{x}$

$\Rightarrow \int \cot \left(v\right)dv=\int \frac{1}{x}dx$

$\Rightarrow ln|\sin \left(v\right)|=|n|x|+k=|n|x|+|nl{e}^k|$

$\Rightarrow ln|\sin \left(v\right)|+ln|e^{k}x|$

$\Rightarrow \sin \left(v\right)=cx$ $\{e^k=c\}$

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