Definition: First-Order Ordinary Differential Equation (ODE)

Let \(U\subseteq \mathbb {R} ^{2}\) be a subset and let $$f\colon\begin{cases} U & \mapsto \mathbb {R},\cr (t,y) & \mapsto f(t,y), \end{cases}$$ be a funktion. The equation

\[\frac{dy(t)}{dt}=f(t,y(t)),\quad\text{short form notation: }y'=f(t,y)\]

is called a first-order ordinary differential equation 1st order ODE) of \(y\).

Please note that the 1st order ODE of \(y\) represents a whole class of many functions \(y\), for which the derivative of \(y\) at \(t\) equals the value of the vector field \(f\) on \(y\) at \(t\).

  1. Proposition: Differential Equation of the Exponential Function

Propositions: 1

Thank you to the contributors under CC BY-SA 4.0!



Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück