$\text{ The solution of the differential equation }\frac{dy}{dx}-y=1-e^{-x}\text{ and }y\left(0\right)=y\text{ , has a finite }$$\text{ value when }\mathrm{x}\to \mathrm{\infty }\text{ then the value of }\left|\frac{2}{y_0}\right|\text{ is }$

$\frac{dy}{dx}-y=1-e^{-x},I.F=e^{-x}$

$\begin{array}{l}\text{ Solution of the D.E is }{\mathrm{ye}}^{-\mathrm{x}}=\int {\mathrm{e}}^{-x}\left(1-{\mathrm{e}}^{-\mathrm{x}}\right)\mathrm{dx}=\int \left({\mathrm{e}}^{{ }^{-x}}-{\mathrm{e}}^{-2\mathrm{x}}\right)\mathrm{dx}\\ \\ y=-e^{-x}+\frac{e^{-2x}}{2}+c\text{ at }x=0\\ \mathrm{y}=-1+\frac{1}{2}+\mathrm{c}\mathrm{y}=-\frac{1}{2}+\mathrm{c}\Rightarrow \mathrm{c}=\mathrm{y}+\frac{1}{2}\\ \therefore {\mathrm{ye}}^{-\mathrm{x}}=-{\mathrm{e}}^{-\mathrm{x}}+\frac{{\mathrm{e}}^{-2\mathrm{x}}}{2}+\mathrm{y}+\frac{1}{2},\text{ if }\mathrm{x}\to \mathrm{\infty }\\ \to 0=0+0+{\mathrm{y}}_0+\frac{1}{2}\to {\mathrm{y}}_0=-\frac{1}{2}\\ \Rightarrow \left|{\mathrm{y}}_0\right|=0.5\end{array}$