Description
Treten in einer Differentialgleichung nur Ableitungen nach einer Variablen (z.B. nur nach Zeit), dann nennt man sie eine gewöhnliche Differentialgleichung. Wir nutzen den Ausdruck, um DGLs von Partielle Differentialgleichung zu unterscheiden.
This page serves as aMoC.
Classification
There are many different differential equations. Some being easily solvable and others aren’t. We can classify them into different types. This allows us to find solutions to whole classes of equations at once. Some simple classifications are:
- Linear ODE and Non-linear differential equation: Linear Differential equations are well-studied and have relatively easy solutions. Non-linear differential equations appear more often in nature and can be very complex.
- Explicit differential equation and Implizite Differentialgleichung: Explicit differential equations describes explicitly how one variable depends on the other variables. An implicit equation is governed by some functional equation where the dependence is not entirely clear.
Especially linear and non-linear differential equations can be sub-classified even further. Then there are some isolated kinds of differential equations with specialised ways of solving them:
- Exact differential equation: A simple implicit equation which consists of a sum which has a potential function.
- First-order ODE: They are typically non-linear but only of first order and dimension 1. We have many different approaches to solves those equations.
- Picard iteration: Not exactly a type of equation. I mean equations that can be solved using this machinery.
Existence and Uniqueness of solutions
The theory behind existence and uniqueness of solutions is described in Existence of ODE solutions and Uniqueness of ODE solutions.