Question

A curve $y=f\left(x\right)$ is passing through $(0,0).$ If the slope of the curve at any point $(x,y)$ is equal to $(x+xy),$ then the number of solution(s) of the equation $f\left(x\right)=1,$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

Given : $\frac{dy}{dx}=x+xy$
$\Rightarrow \frac{dy}{(1+y)}=xdx$
$\Rightarrow \ln |1+y|+\ln C=\frac{x^2}{2}$
$\Rightarrow C(1+y)=e^{x^2/2}(\because e^{x^2/2}>0)$
Since $y=f\left(x\right)$ passes through $(0,0),$
$\therefore C(1+0)=1\Rightarrow C=1$
$\therefore y=e^{x^2/2}-1$
For $f\left(x\right)=1,$ we have
$e^{x^2/2}=2$


$\therefore$ Number of solutions is $2.$