Question

The solution of the differential equation $ydx-xdy=y^2\tan \left(\frac{x}{y}\right)dx$ is
( $C$ is constant of integration)

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

Answer 1

The correct option is B $\sin \left(\frac{x}{y}\right)=C{e}^x$

$ydx-xdy=y^2\tan \left(\frac{x}{y}\right)dx$
$\Rightarrow \cot \frac{x}{y}\left(\frac{ydx-xdy}{y^2}\right)=dx$
$\Rightarrow \cot \left(\frac{x}{y}\right)d\left(\frac{x}{y}\right)=dx$
Integrating both sides, we get
$\log \left(\sin \frac{x}{y}\right)=x+\log C$
$\Rightarrow \sin \frac{x}{y}=e^{x+\log C}=C{e}^x$