> [!NOTE] Lemma > > Let $n\in \mathbb{N}^{+}.$ Any [[Scalar Ordinary Differential Equation]] of order $n$ can be transformed to [[System of First Order Ordinary Differential Equations|n x n System of First Order Ordinary Differential Equations]]. **Proof**: For $1\leq i \leq n,$ let $x_{i}(t)=x^{(i-1)}(t).$ Then $x_{i}'(t)-x_{i+1}(t)=0$ for all $1\leq i\leq n-1.$ Also $F(t,x_{1}(t),x_{2}(t),..,x_{n}'(t))=0.$There are $n$ equations in total involving $n$ dependent variables and each equation is of order $1.$