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For $U=A{x}^m$, $\psi =\sqrt{\nu xU}f(\eta )$ giveswith $f(0)=f^{\prime }(0)=0$ and $f^{\prime }(\infty )=1$.

This equation and its boundary conditions determine the Falkner-Skan boundary layer.

The second Painlevé equation, in its standard normalization. Similarity reductions of some integrable systems give equations of this form, after rescaling variables.