Differential Equations

My ODE

ODE’s

Fundamentals

In nature, systems are more naturally characterized by their rates of change than by closed-form functions.
Usually more interested in quality aspects e.g. the asymptotic behaviour of the solution for $t\to \infty$, the stability of fixed points or bifurcations in the parameter domain

ODE definition

Def

An ordinary differential equation of order $n$ is an equation for the unknown $y=y(x)$ and its derivative(s)
Implicit form : $F(x,y^{\prime },y^{\prime \prime },\dots y^n)=0$
Explicit form:
$y^{(n)}=G(x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n-1)})$
($F:\Omega \to \mathbb{R}$, a continuous function with region $\Omega \subseteq I\times {\mathbb{R}}^{n+1}$ and interval of validity $I\subseteq \mathbb{R}$)

ODE solution

The set of all solutions of an ODE is called the general solution

Def

The solution of a ODE is an N-times differentiable function $y=y(x)$ for which the ODE is valid for any $x\in I$

Warning

It’s possible that the ODE is not defined for all $x\in \mathbb{R}$ : e.g. $y^{\prime }=-\frac{y}{x}$

Initial Value Problem (IVP)

To find a particular solution, n constrains must be defined

Form

$$\{\begin{array}{rl}F(x,y,y^{\prime },y^{\prime \prime },\dots ,y^{(n)})& =0,(x,y,\dots ,y^{(n)})\in \Omega \\ y(x_0)& =y_0\\ y^{\prime }(x_0)& =y_1\\ ⋮& \\ y^{(n-1)}(x_0)& =y_{n-1}\end{array}$$

$(x_0,y_0,y_1,\dots ,y_{(n-1)}\in \mathbb{R})$

Existence and uniqueness of solutions

Def

f is Lipschitz-continuous with respect to $y$, if

$$|f(x,y_1)-f(x,y_2))|\le L|y_1-y_2|$$

For $L$ a constant not depending on $y$
→ It’s the same as saying $f$ is continuous on $\Omega$ and $\frac{\partial f(x,y)}{\partial y}$ is continuous in both x and y on $\Omega$ which is sometimes easier to check

Classification

Slope fields

System of ODE’s

Difference equations

Existence of solutions

Analytical methods for First Order ODEs

Separable Equations

$y^{\prime }=g(x)h(y)$

Linear Equations

$y^{\prime }+p(x)y=q(x)$

Homogeneous vs Inhomogeneous

Integrating Factor method

$y^{\prime }=\frac{g(x)}{h(y)}$

Graphical Methods **

Direction Fields

Orthogonal Trajectories

Nonlinear

Overview

Special cases

Analytical methods for Second Order ODEs

Homogeneous Linear Equations (Constant Coefficients)

Characteristic Equation

$a{r}^2+br+c=0$

Cases

Real distinct roots, Repeated roots, Complex roots

Inhomogeneous Linear Equations

Principle of Superposition

Method of Undetermined Coefficients (Ansatz method)

Variation of Parameters

System of ODEs

Analytical methods

Oscillation problems

Numerical Methods

Taylor series expansion

Euler’s Method

Heun’s Method (Improved Euler)

Runge-Kutta Method (RK4)

Error, convergence and stability

Dynamical systems

One-dimensional continuous systems

Two-dimensional continuous systems

Three-dimensional continuous systems and chaos

Discrete dynamical systems

Handwritten