> [!NOTE] Definition 1 (Linear Scalar Ordinary Differential Equation) > Let $n\in \mathbb{N}^{+}.$ Let $F(t,x(t),x'(t), \dots , x^{(n)}(t))=0$be an $n$th order [[Scalar Ordinary Differential Equation|scalar ordinary differential equation]] with $F:(\alpha,\beta)\times \mathbb{R}^{n+1}\to \mathbb{R}.$ > >Define $s:(\alpha,\beta)\to \mathbb{R}$ by $s(t)=F(t,0,0,\dots,0).$The equation is homogenous iff $s=0.$