If d y   d x = 2 x y + 2 y · 2 x 2 x + 2 x + y log e 2 ,   y 0 = 0 , then for y = 1 , the…

Given differential equation is

$\frac{dy}{dx}=\frac{2^x·y+2^y·2^x}{2^x+2^{x+y}{\log }_{e}2}$

$\Rightarrow \frac{dy}{dx}=\frac{2^x\left(y+2^y\right)}{2^x\left(1+2^y{\log }_{e}2\right)}$

$\Rightarrow \int \frac{1+2^y{\log }_{e}2}{y+2^y}dy=\int dx$

$\Rightarrow \int \frac{d\left(y+2^y\right)}{y+2^y}=\int dx$

$\Rightarrow \ln \left|y+2^y\right|=x+C\left[\because \int \frac{f^{\text{'}}\left(x\right)}{f\left(x\right)}dx=\ln \left(f\left(x\right)\right)+C\right]$

Now $\because y\left(0\right)=0$

$\Rightarrow C=0$

$\therefore \ln \left|y+2^y\right|=x$

Now for $y=1$ we have

$x=\ln \left(1+2\right)=\ln 3\in \left(1, 2\right)$