= Autonomous third-order reduction using the dependent variable
{title2=$f'=G(f),\quad f''=GG_f,\quad f'''=G^2G_{ff}+G(G_f)^2$}
An <autonomous differential equation> for $f(\eta)$ can be reduced by one order on a monotone solution branch by using $f$ as the new independent variable. Setting $G(f)=df/d\eta$ and applying the <chain rule> gives the displayed <derivative> identities. The absence of explicit $\eta$ dependence makes the resulting equation involve only $f,G$ and <derivatives> of $G$. A point with $G=0$ is singular for this coordinate change; for example $f\sim b\eta^2/2$ gives $G\sim\sqrt{2bf}$ on an appropriate branch, so $G_f$ need not remain finite there.
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