'Go with the flow'

Ordinary differential equations: Slope field and solution curves

'Go with the flow'

A slope field of a differential equation $\dfrac{\dd y}{\dd t}=\varphi(t, y)$ can give a good impression of how the solution curves look like because these graphs should always touch lineal elements. By drawing a smooth curve for which any point ion the curve is tangent to the lineal element associated with that point you get a so-called integral curve; the graph therefore is a graphical solution of the differential equation. Such a curve has already been referred to as solution curve. We consider two simple examples.

Example 1 We consider the differential equation

$\frac{\dd y}{\dd t}=2$ In the figure below, we added to the slope field graphs of six solutions, namely of $y=2t+c$ with, from left to right $c=3, 1.5, 0.1, -2.5, 3.75, -6$ . All solutions are line graphs with slope $1$ because all lineal elements are parallel and have slope $1$ . You can also get a solution of the differential equation by always going in the direction of lineal elements.

Example 2 In the figure below you see on the left-hand side the slope field of the differential equation

$\frac{\dd y}{\dd t}=t$ and on the right-hand side the same slope field but now with graphs of solutions, namely those of $y=\tfrac{1}{2}t^2+c$ with different values of $c$ . Note that the solution curves always go in the direction of lineal elements; This is what we mean when we write 'go with the flow'.

In this slope field, the isoclines are vertical lines which are steeper when the value $t$ differs more from $0$ . The nullcline is the vertical axis: this separates lineal elements with negative and positive slopes.