I do not find how the following system equation is formulated. The following example is about the Deterministic Continuous-Time Optimal Control chapter.
We are considering a continuous-time dynamic system $$\dot{x}(t) = f(x(t),u(t)), \qquad 0\le t\lt T, \tag1\\ x(0) : \text{given,}$$ where $x(t)\in\Bbb{R}^n$ is the state vector at time $t$, $\dot{x}(t)\in\Bbb{R}^n$ is the vector of first order time derivatives of the states at time $t$, $u(t)\in U \subset \Bbb{R}^m$ is the control vector at time $t$, $U$ is the control constraint set, and $T$ is the terminal time.
The system $(1)$ represents the $n$ first order differential equations $$\frac{dx_i(t)}{dt} = f_i(x(t), u(t)), \qquad i=1, \cdots, n.$$
Example 3.1.2 (Resource Allocation)
A producer with production rate $x(t)$ at time $t$ may allocate a portion $u(t)$ of his/her production rate to reinvestment and $1-u(t)$ to production of a storable good. Thus $x(t)$ evolves according to $$\dot{x}(t)=\gamma u(t)x(t),\tag2$$ where $\gamma\gt0$ is a given constant. The producer wants to maximize the total amount of product stored $$\int_0^T \left(1-u(t)\right)x(t)dt$$ subject to $$ 0\le u(t)\le 1, \qquad \text{for all } t\in [0, T].$$ The initial production rate $x(0)$ is a given positive number.
I do not understand where and how the equation $(2)$ is derived from. Please let me understand it. Thank you for reading my question.