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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?

  • 0
    are there any boundary values specified for your equation?2012-07-11
  • 0
    Woops!! Yes there are!2012-07-12

2 Answers 2