Ordinary Differential Equations

Definition: Ordinary Differential Equation

An ordinary differential equation of order $n$ is an equation which can be written as

$$F\left(x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)}\right)=0,$$

where $F$ is some expression containing $x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)}$.

Definition: Solution to an ODE

A solution to the ODE on the subset $\mathcal{D}\subseteq \mathbb{R}$ is an $n$-times differentiable real function $\varphi :\mathcal{D}\to \mathbb{R}$ such that

$$F\left(x,\varphi (x),{\varphi }^{\prime }(x),{\varphi }^{\prime \prime }(x),\dots ,{\varphi }^{(n)}(x)\right)=0\forall x\in \mathcal{D}$$

Ordinary Initial Value Problems

Definition: Ordinary Initial Value Problem

An ordinary initial value problem of order $n$ consists of an $n$-th order ODE

$$F\left(x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)}\right)=0$$

and a system of $n$