A direction field. [Picture]

Figure 1: A direction field (aka a slope field)

An ordinary differential equation (abbreviation \(``\)de") is an equation involving derivatives of an unknown function (usually \(y\)) of one variable (usually \(x\) or \(t\)).

A solution to a de is a function that satisfies the de on some open interval,

The graph of a solution is called a solution curve of the de. Any curve made of solution curves of a de is called an integral curve of the de.

For example, \(y=\sin {(t)}\) is a solution to the de \(y''=-y\) on \((-\infty ,\infty )\).

For another example, \(y=-\frac {1}{x}\) is a solution to \(y'=y^2\) but only on open intervals not containing \(x=0\).

The order of a de is the order of the highest-order derivative involved.

For example, the order of \(y''=\frac {t^2}{y'}\) is \(2\).

Any first order equation can be solved for \(y'=f(x,y)\) (or
\(y'=f(t,y)\)).

If \(f\) is defined on a region \(R\) in the \(xy\)-plane then the direction field (aka slope field) for the de \(y'=f(x,y)\) on \(R\) is a graph consisting of short line segments (or arrows) with slope determined by \(f(x,y)\) at each \((x,y)\) in \(R\).