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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 \(t\).

Table of Contents

1. Proposition: Differential Equation of the Exponential Function

Mentioned in:

Propositions: 1

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References

Adapted from CC BY-SA 3.0 Sources:

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