Question

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

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

Answer 1

Answer: (A)

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