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Proposition: Suppose that $f:\mathbb{R^n} \rightarrow \mathbb{R^n}, g:\mathbb{R^n} \rightarrow \mathbb{R}$ is a positive function, and $\phi$ is the flow of the differential equation $\dot{x}=f(x)$. If the family of solutions of the family of initial value problems $$\dot{y} = g(\phi(y,\xi)),$$ $$y(0)=0,$$ with parameter $\xi \in \mathbb{R^n}$, is given by $\rho: \mathbb{R} \times \mathbb{R^n} \rightarrow \mathbb{R}$, then $\psi$, defined by $\psi(t,\xi)=\phi(\rho(t,\xi),\xi)$ is the flow of the differential equation $\dot{x}=g(x)f(x)$.

I have crossposted this here as well: differential equations - Interpreting another proposition full of symbols - Mathematics Stack Exchange