First of all a clarification of definition: A continuous time dynamical system in this question is defined as the set $\mathbb{R}^n$, together with a function $W^t(x): \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $W^t(x)$ satisfies $W^t(x) \circ W^s(x) = W^{t+s}(x)$. A flow of the system is a function $f(t): \mathbb{R} \rightarrow \mathbb{R}^n$ such that there exists some $x_0 \in X$ such that $f(t) = W^t(x_0)$ for all $t$.
Now given a dynamical system $(W^t(x), \mathbb{R}^n)$ specified by some vector field $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ (I understand this statement as saying $\frac{d}{dt} W^t(x) = F(x)$ where $x$ can be seen as a function of $t$), I want to show that some function $\phi(t)$ is a flow of the system given that $\phi'(t) = F(\phi(t))$. I am trying to prove this directly using the definition of continuous time dynamical system given above but I got stuck.
To give an example, I want to show that $\phi(t) = (-e^{-2t}, \sqrt{3}e^{-t})$ is a flow in the dynamical system $(W^t, \mathbb{R}^2)$ specified by the vector field $F(x,y) = (x+y^2, -y)$. I can show that $\phi(t)$ is a solution to the system of equation $x' = x+y^2, y' = -y$. But how can I use this fact to prove that $\phi$ is a flow of the dynamical system using the above definition? Is it possible that we find a point $x_0 \in \mathbb{R}^2$ and show that $\phi(t) = W^t(x_0)$? Also can this be proven without solving for $W^t$?
I hope this question makes sense since I still have some trouble understanding the concept of a continuous dynamical system.