• # question_answer A curve $y=f(x)$ 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 of the equation $f(x)=1$, is A)  0                     B)  1 C)  2                                 D)  4

$\frac{dy}{dx}=x+xy\Rightarrow \,\frac{dy}{dx}-xy=x$ Integrating factor $={{e}^{\frac{-{{x}^{2}}}{2}}}$ $\therefore \,\,y.{{e}^{\frac{-{{x}^{2}}}{2}}}\,=\int_{{}}^{{}}{x{{e}^{\frac{-{{x}^{2}}}{2}}}\,dx=-{{e}^{\frac{-{{x}^{2}}}{2}}}+C}$ $\Rightarrow \,y=C.{{e}^{\frac{{{x}^{2}}}{2}}}\,-1$ At $x=0,\,\,y=0\Rightarrow \,C=1$ $\therefore \,\,f(x)={{e}^{\frac{{{x}^{2}}}{2}}}\,-1$ So, $f(x)=1\Rightarrow \,{{e}^{\frac{{{x}^{2}}}{2}}}=2$ $\Rightarrow$ Number of solution is 2.