I'm given $\epsilon y'=y=e^{-t}$. I went through and found the outer approximation and then proceeded to find the inner approx. I rescaled and balanced the equation to get: $Y'+Y=e^{- \tau\epsilon}$ where I've defined $\tau=\frac{t}{\delta(\epsilon)}$
My book shows that this is: $Y'+Y=1$ and then somehow gets to the solution being $Y_i(\tau)= 1+Ce^{{t}/{\epsilon}}$
I can't seem to figure out how they got that. If the assumption holds that $\epsilon=0$, than how does the characteristic polynomial $r+1=1$ yield that $Y_i(\tau)$?
The mistake is in the characteristic polynomial. It needs to be set equal to $0$ and then solve for the homogeneous. From there you calculate the particular solution using an integrating factor. The answer is fairly easy from there. $y_p=1$ and $y_h=e^{-\tau}$