Question

If $c$ is any arbitrary constant, then the general solution of differential equation $ydx-xdy=xydx$ is given by -

• A. $y=cx{e}^x$
• B. $y=cx{e}^{-x}$
• C. $y+e^x=cx$
• D. $y{e}^x=cx$

Answer 1

Answer: (B)

$y=cx{e}^{-x}$

$ydx-xdy=xydx$

$ydx-xydx=xdy$

$y(1-x)dx=xdy$

$\frac{1-x}{x}dx=\frac{dy}{y}$

Integrating both side

$\int \left(\frac{1-x}{x}\right)dx=\int \frac{dy}{y}$

${\log }_{e}x-x={\log }_{e}y+c$

${\log }_e\frac{x}{y}=x+c$

$\frac{x}{y}=e^c\cdot e^x$

$y=e^{c}x{e}^{-x}$

suppose $c=e^c$

then,

$y=cx{e}^{-x}$