6.1. Definition

Many dynamic processes can occur simultaneously, giving more than one ODE. For example, consider two tanks. Tank 1 has concentration \(c_1\) of a solute and Tank 2 has concentration \(c_2\). Tank 1 flows into Tank 2, which is then passed through a pump and recycled back to Tank 1.

Eventually, the two concentrations should equalize, but how do they evolve to over time? \(c_1\) and \(c_2\) can be modeled using unsteady mass balances. This is a system of first-order ODEs that can be solved simultaneously.

The general, explicit form of a system of first-order ODEs is:

(6.1)\[\begin{align} y_1' &= f_1(t, y_1, \cdots, y_n)\\ \vdots \\ y_n' &= f_n(t, y_1, \cdots, y_n) \end{align}\]

where t is the independent variable, \((y_1, \cdots, y_n)\) are the n dependent variables, and \((f_1, \cdots, f_n)\) are the n right-hand side functions of the n ODEs. We will notate this system more compactly using vectors

(6.2)\[\begin{equation} \vv{y}' = \vv{f}(t, \vv{y}) \end{equation}\]

where y is the column vector of dependent variables, f is the column vector of right-hand side functions, and \(\vv{y}'\) is a shorthand for the column vector of first derivatives of all dependent variables.