How to decide whether it is a system of Differential Algebraic Equations or a System of Ordinary Differential Equations?

Question

I am struggling to name some of my dynamic models right. To be specific, I am not sure whether I should call it a system of Differential Algebraic Equations (DAEs) or a System of Ordinary Differential Equations (ODEs) with parameters defined by algebraic equations.

Example:

Equations:

$\frac{dx_1}{dt} = x_1 \times (x_2 \times a_0)$

$\frac{dx_2}{dt} = x_2 + (x_1 \times a_1)$

$a_0 = c_0 \times T$

$a_1 = c_1 \times T$

$T = c_2 \times t$

$c_0, c_1, c_2$ are constants $\in \mathbb{R} $

$a_0, a_1, T, x_1, x_2$ are variables $\in \mathbb{R} $

Is it save to define such systems as DAEs?

Answer 1

You can call it a DAE, but since it is so easy to replace both $a_0$ and $a_1$ by a constant times $t$, better do it first and than call the result a differential equation.