Non-linear Ordinary Differential Equations and Dynamical Systems, Introduction to

An ordinary differential equation (ODE) is called linear if it can be written in the form $d y / \text{d} x = f(x)y+g(x)$ with x a real or complex variable and y an n‑dimensional (finite) real or complex vector function. Non‐linear ODEs are then n‑dimensional ODEs of the form $\text{d} y / \text{d} x = F(x,y)$that are not linear. Jean Mawhin famously compared this distinction to a division of the animal world into ‘elephants’ and ‘non‐elephants’, but even admitting that the distinction between linear and non‐linear ODEs is artificial and of relatively recent date, it makes a little bit more sense than it looks like at first sight. The reason is, that in many problem formulations in classical physics, linear equations are quite common. Also, when considering non‐linear ODEs, but linearizing around particular solutions, a number of fundamental theorems can be used to characterize the particular solutions starting with the features of the linearized equation. This will become...