I am in the midst of solving this equation
$\epsilon \ddot{y}+\dot{y}+1-\frac{1}{(y+1)^{2}}=0$ with the boundary condition $y(0)=1$ and $\dot{y}(0)=-1$
and $\epsilon$ is small. To start off with, I asymptotically expanded $y$ to yield
$y=y_{0}+\epsilon y_{1}$
and substituted the above and solve for $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ for the outer solution. The thing is, how can we make sure that the asymptotic expansion chosen for $y$ is correct? What happens if I choose to expand $y$ like
$y=y_{0}+\epsilon ^2 y_{1}$ or even $y=y_{0}+\sqrt{\epsilon} y_{1}$ to solve the outer solution?