If $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2^{\mathrm{x}} \mathrm{y}+2^{\mathrm{y}} \cdot 2^{\mathrm{x}}}{2^{\mathrm{x}}+2^{\mathrm{x}+\mathrm{y}} \log _{e} 2}, \mathrm{y}(0)=0$, then for $\mathrm{y}=1$
Correct Option: 1
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2^{\mathrm{x}}\left(\mathrm{y}+2^{\mathrm{y}}\right)}{2^{\mathrm{x}}\left(1+2^{\mathrm{y}} \ell \mathrm{n} 2\right)}$
$\Rightarrow \int \frac{\left(1+2^{y}\right) \ln 2}{\left(\mathrm{y}+2^{y}\right)} \mathrm{dy}=\int \mathrm{dx}$
$\Rightarrow \ell \mathrm{n}\left|\mathrm{y}+2^{y}\right|=\mathrm{x}+\mathrm{c}$
$x=0 ; y=0 \Rightarrow c=0$
$\Rightarrow x=\ell \mathrm{n}\left|y+2^{\prime}\right|$
$\Rightarrow$ at $y=1, x=\ell \mathrm{n} 3$
$\because 3 \in\left(\mathrm{e}, \mathrm{e}^{2}\right) \Rightarrow \mathrm{x} \in(1,2)$
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