Question:
The solution curve of the differential equation,
$\left(1+\mathrm{e}^{-\mathrm{x}}\right)\left(1+\mathrm{y}^{2}\right) \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}^{2}$, which passes
through the point $(0,1)$, is :
Correct Option: 1
Solution:
$\left(1+\mathrm{e}^{-\mathrm{x}}\right)\left(1+\mathrm{y}^{2}\right) \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}^{2}$
$\Rightarrow\left(1+y^{-2}\right) d y=\left(\frac{e^{x}}{1+e^{x}}\right) d x$
$\Rightarrow\left(y-\frac{1}{y}\right)=\ln \left(1+e^{x}\right)+c$
$\therefore \quad$ It passes through $(0,1) \Rightarrow \mathrm{c}=-\ell \mathrm{n} 2$
$\Rightarrow \quad \mathrm{y}^{2}=1+\mathrm{y} \ell \mathrm{n}\left(\frac{1+\mathrm{e}^{\mathrm{x}}}{2}\right)$