> [!NOTE] Definition (System of First Order Ordinary Differential Equations) > Let $n\in \mathbb{N}^{+}.$ A $n\times n$ system of ordinary differential equations is a [[System of Differential Equations|system]] of the form $\underline{F}(t,\underline{x}(t),\underline{x}'(t))=\underline{0} \tag{1}$for some [[Vector Valued Function of Several Real Variables|vector valued function]] $\underline{F}:(\alpha,\beta)\times \mathbb{R}^{n} \times \mathbb{R}^{n}\to \mathbb{R}^{n}$ where $n$ is the number of dependent variables. > # Properties **Explicit form**: equation $(1)$ is explicit if it can be rewritten in the form $\underline{x}'(t)=\underline{f}(t,\underline{x}(t))$ where $\underline{f}:(\alpha,\beta)\times\mathbb{R}^{n}\to \mathbb{R}^{n}.$ **Classification**: A system of first-order equations is autonomous if every component equation is autonomous: that is none of the derivatives depend on the independent variable. they have stationary points. Linear systems **Visualisation**: Note that [[2 x 2 System of First Order Ordinary Differential Equations|2 x 2 systems]] can be visualised by [[Phase Portrait|phase portraits]] and [[Direction Field|direction fields]]. **Solution**: .... # Applications [[Reduction of Order of Scalar Ordinary Differential Equations]]. [[Homogenous Linear 2 x 2 System of First Order Ordinary Differential Equations with Real Coefficients]]