Differential equation and solutions

Question

If $y(x)$, which is a solution of differential equation $\dfrac{dy}{dx}-y=1-e^{-x}$ and $y(0)=a$, has a finite value as $x\to \infty$, then find $a$.

Answer 1

Take $y=w+ae^{-x}+b$ therefore:$$y'-y=w'-w-ae^{-x}-ae^{-x}-b=1-e^{-x}\to\\b=-1,a=\frac{1}{2}$$also $w$ is the response to differential equation $w'=w$ menawhile $w=Ce^x$ therefore we obtain $$y=Ce^x+\frac{1}{2}e^{-x}-1$$ This answer has a finite value for $x\to\infty$ iff $C=0$ which leads to $a=y(0)=-\frac{1}{2}$