> [!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 $F:(\alpha,\beta)\times \mathbb{R}^{n+1}\to \mathbb{R}.$ > > Then the equation is linear iff $F$ can be rewritten as $F(t,x(t),x'(t),\dots,x^{(n)}(t)) = s(t) + \sum_{i=0}^{n} a_{i}(t) x^{(i)}(t)$with some functions $s,a_{i}:(\alpha,\beta)\to \mathbb{R},$ $i=0,1,\dots,k.$ **Note**: $s(t) = F(t,0,0,\dots,0).$