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Asymptotic behavior of functions, asymptotic series and related topics

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Asymptotic solution of a system of ODEs

But I still do not know how to get asymptotics. The remaining equations are $$ 2 a'(u)=u x'(u),\\ 4 a(u)^2 \left(3 u^2 x(u)^2+1\right)=4 x(u) \big(u a(u)+4 x(u)\big)+1. $$ …
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