Consider a differential equation for a dependent variable \(y\) as a function of \(t\) where the rate formula might depend on both \(y\) and \(t\). Mathematically, we would write this \begin{equation*}\frac{dy}{dt} = f(t,y),\end{equation*} a function of both \(t\) and \(y\). This differential equation defines a slope at every point in the \((t,y)\) plane.

A slope field is a graphical representation, showing short line segments or vectors at every point of a grid within the plane. To get a computer to draw a slope field, we will need to define the function giving the slopes (the rate function) as well as define the grid on which we perform the computation.

For our example, we will look at the differential equation defined by \begin{equation*}\frac{dy}{dt} = ty-y^2.\end{equation*}

For an example that shows a solution trajectory on top of the slope field, see Section 4.3.