Autonomous third-order reduction using the dependent variable ($f^{\prime }=G(f),f^{\prime \prime }=G{G}_f,f^{\prime \prime \prime }=G^2{G}_{ff}+G(G_f{)}^2$)

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.