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Regarding matched asymptotic expansions, given an ODE BVP, I have learned a boundary layer can arise, where we need to stretch the independent variable through carefully selection i.e. if ##x## is the independent variable, perhaps ##\phi = (1-x)/\epsilon : \epsilon \ll 1##.

Would we ever see a situation where we would have to stretch the dependent variable, say the ODE was over ##y(x)##. Something perhaps like ##Y = y/\epsilon : \epsilon \ll 1##, as an example.

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# A Matched Asymptotic Expansion and stretching variables

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