All variables of a system constitute its phase space. A selection of a few variables is referred to as a phase space projection. For a differential equation are three meaningful projections of the phase space.
The general characteristics divergent/ convergent/ oscillating of a differential equation, can be visualized well in a diagram that shows, in addition to the function and its derivative also the projection .
In Fig.5.9 the phase space projection for the system with the differential equation is shown in the right window. The adjustable constant determines the number of periods in the interval
For the case of the trigonometric function is for a circle that is transversed periodically; for the curve becomes because of the factor an ellipse and is not closed (why?). For this ellipse is transversed multiple times.
In this case the differential equation is particularly simple. More complex differential equations of order define families of more involved functions. One can however always differentiate between solutions that converge, diverge or oscillate with increasing variables, and the phase space projections make this difference particularly apparent.
Later we will visualize solutions of differential equations in more detail.