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 \(t\).
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