> [!NOTE] Definition > A 2 x 2 System of First Order Ordinary Differential Equations is a [[System of First Order Ordinary Differential Equations|system of first order equations]] of the form $\underline{F}(t,\underline{x}(t),\underline{x}'(t))=\underline{0}$where $\underline{F}:(\alpha,\beta)\times \mathbb{R}^{2} \times \mathbb{R}^{2}\to \mathbb{R}^{2}.$ # Properties **Explicit form**: in some cases, we can rearrange the system to the explicit form $\begin{align} x_{1}'(t)= f_{1}(t, x_{1}(t),x_{2}(t)) \\ x_{2}'(t)= f_{2}(t, x_{1} (t), x_{2}(t)) \end{align}$ where $f_{1},f_{2}:(\alpha,\beta)\times\mathbb{R}^{2}\to \mathbb{R}.$ A has the form $\underline{x}'(t)=A\underline{x}(t).$ # Applications **Examples**: ......