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I encountered the following ODE and tried to solve using perturbation theory: $$y'=(1+\frac{1}{100x^2})y^2-2y+1$$ $$y(1)=1,\ x\in[0,1]$$ I am asked to find an approximation correct to $O(\epsilon)$. Next I need to Introduce a Boundary-Layer-type variable ($X=\frac{x}{\epsilon}$) and find the layer solution valid to $O(1)$. So I dont know how to define a proper $\epsilon$ in the question. And at which side I am suppose to define the boundary layer solution.

I am rather lost since it differs from everything i've seen in the subject.

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  • $\begingroup$ Finding a solution to $O(\epsilon)$, where $\epsilon$ is not defined, is definitely a pretty vague question. Do you have any more information? $\endgroup$ – David Jun 9 '16 at 1:52
  • $\begingroup$ I am suppose to define the small parameter myself. Thought about defining it by $\epsilon = \frac{1}{100}$ and then look for a valid domain of the solution, but I am really not sure. $\endgroup$ – yoni Jun 9 '16 at 6:47
  • $\begingroup$ That seems like a good idea to me. Roughly, as long as $x$ is larger than $0.1$, then the $(10x)^{-2}$ term is small. The boundary layer should be at $x=0$, which you have hinted at in your question, because this is where that terms is not small. $\endgroup$ – David Jun 9 '16 at 6:51
  • $\begingroup$ Equal question: math.stackexchange.com/q/1269999/115115 $\endgroup$ – Dr. Lutz Lehmann Sep 12 '16 at 10:34

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