Definition of a Linear Ordinary Differential Equation

Any $n$th ordinary differential equation (ODE) can be written as $F(x,y,y^{prime},\dots ,y^{(n)})=0$. The ODE is linear if it can be written in the form…

$$a_n(x)\frac{d^{n}y}{d{x}^n}+\dots +a_1(x)\frac{dy}{dx}+a_0(x)y=g(x)$$