# Interpreting of another proposition full of symbols

#### kalish

##### Member
Could someone help me interpret the following proposition full of symbols? I've been struggling to comprehend it as well. Thanks in advance.

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

#### HallsofIvy

##### Well-known member
MHB Math Helper
Could someone help me interpret the following proposition full of symbols? I've been struggling to comprehend it as well. Thanks in advance.

Proposition: Suppose that $f:\mathbb{R^n} \rightarrow \mathbb{R^n}, g:\mathbb{R^n} \rightarrow \mathbb{R}$ is a positive function,
f is a function that maps n real variables to a value of n ordered numbers (OR f maps an n dimensional vector to an n dimensional vector). g is a function that maps n real variables to a single positive number (OR g maps an n dimensional vector to a single number).

and $\phi$ is the flow of the differential equation $\dot{x}=f(x)$.
$$\phi$$ is the general solution to the differential equation. Note that the equation can be thought of as a set of n interrelated differential equations.

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