Question

The slope of a curve at any point $(x,y)$ other than the origin, is $y+\frac{y}{x}$. Then, the equation of the curve is

• A. $y=Cx{e}^x$
• B. $y=x(e^x+C)$
• C. $xy=C{e}^x$
• D. $y+x{e}^x=C$
• E. $(y-x)e^x=C$

Answer 1

Answer: (D)

$y+x{e}^x=C$

Slope of a curve at $(x,y)$ is given by $\frac{dy}{dx}$.
According to the equation,
$\frac{dy}{dx}=y+\frac{y}{x}$
$\Rightarrow \frac{dy}{dx}=y(1+\frac{1}{x})$
$\Rightarrow \int \frac{dy}{y}=\int (1+\frac{1}{x})dx$
$\Rightarrow \log y=x+\log x+\log C$
$\Rightarrow \log y=\log \left(Cx{e}^x\right)$ $[\because x=\log e^x]$
$\Rightarrow y=Cx{e}^x$