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How does one do an asymptotic power expansion of an implicit function? Say the function y...

How does one do an asymptotic power expansion of an implicit function? Say the function y = y(x) is given implicitly as y = 1+x*y^2. If that function is too difficult, then how about x^2+y^2 = 1? I just don't know how to get started. I've thought about using the Taylor series expansion, but I don't know if that is a correct approach.

I was asked to do the power expansion of y = 1+x*y^2 to the power 8. If someone can help doing the power expansion of the first few terms, that would be helpful in getting started. Thank you.

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