Question:
The solution curve of the differential equation,
$\left(1+e^{-x}\right)\left(1+y^{2}\right) \frac{d y}{d x}=y^{2}$, which passes through the point $(0,1)$, is :
Correct Option: , 3
Solution:
$\int\left(\frac{y^{2}+1}{y^{2}}\right) d y=\int \frac{e^{x} d x}{e^{x}+1}$
$\Rightarrow y-\frac{1}{y}=\log _{e}\left|e^{x}+1\right|+c$
$\because$ Passes through $(0,1)$.
$\therefore c=-\log _{e} 2$
$\Rightarrow y^{2}-1=y \log _{e}\left(\frac{e^{x}+1}{2}\right)$
$\Rightarrow y^{2}=1+y \log _{e}\left(\frac{e^{x}+1}{2}\right)$