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        

Correct Answer: C

Solution :

\[\frac{dy}{dx}=x+xy\Rightarrow \,\frac{dy}{dx}-xy=x\] 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.