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

Answer 1

Answer: (A)

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