> [!NOTE] Lemma > Let $F(t,x(t),x'(t),x''(t))=0$be a second order [[Scalar Ordinary Differential Equation|scalar differential equation]]. > > Let $x_{1}(t)=x(t)$ and $x_{2}(t)=x'(t).$ Then they satisfy the [[2 x 2 System of First Order Ordinary Differential Equations|2 x 2 system]] $\begin{align} > x_{1}'(t)-x_{2}(t)=0 \\ > F(t,x_{1}(t),x_{2}(t),x_{2}''(t))=0 > \end{align}$ # Applications **Generalisation**: [[Reduction of Order of Scalar Ordinary Differential Equations]]