The concept of an equilibrium is that the state of the system is not changing. For variables, this means that the dynamic variables are constant functions with respect to time. Constant functions have a zero rate of change. We find equilibria solutions to a differential equation \(\frac{dy}{dt} = F(t,y)\) by solving the equation \(F(t,y) = 0\) and look for solutions of the form \(y=C\) for some constant \(C\). Other solutions that involve the variable \(t\) are not equilibrium solution but instead describe other points where the instantaneous rate of change is zero but for which the function changes at other times.

When we wanted to understand the stability of equilibria resulting from projection function models, we started by considering linear projection functions. For differential equations, we do similar analysis. We start with proportional rates of change models and then apply that to linear rates of change models.

A differential equation of the form \(\frac{dy}{dt} = \alpha y + \beta\) can be transformed into an exponential growth/decay model by rewriting the model in terms of the equilibrium, found by \(\alpha y + \beta = 0\). The equilibrium is \(y^*=-\frac{\beta}{\alpha}\) so that the growth rate model can be rewritten as \begin{equation*} \frac{dy}{dt} = \alpha(y - y^*). \end{equation*} Because \(y^*\) is a constant, \(\frac{dy^*}{dt} = 0\). The rules of calculus allow us to define \(u = y - y^*\) and find the differential equation \begin{equation*}\frac{du}{dt} = \frac{d}{dt}[y-y^*] = \frac{dy}{dt} = \alpha (y-y^*) = \alpha u. \end{equation*} Thus, \(u\) is an exponential model and \(y\) is shifted to be centered around the equilibrium.

Nonlinear growth rates that depend only on the dependent variable, \(\frac{dy}{dt} = f(y)\), have equilibria at all of the values \(y\) so that \(f(y)=0\). Consider an equilibrium at \(y=y^*\). We can approximate the differential equation \(\frac{dy}{dt} = f(y)\) by the tangent line approximation at \(y=y^*\) to determine stability of the equilibrium as long as \(f'(y^*) \ne 0\). Then approximation suggests \(f(y) \approx f'(y^*) \cdot (y-y^*)\) for values of \(y\) sufficiently near \(y^*\). This is a linear rate function that leads to a shifted exponential model.

A differential equation with a rate that depends only on the dependent variable is called autonomous, such as \(\frac{dy}{dt} = f(y)\). This is analogous to a sequence that can be defined recursively with a projection function. For a recursive sequence, we could study it geometrically using a cobweb diagram. When the projection function is above the line \(y=x\), the sequence increases; when the projection function is below the line, the sequence decreases.

For an autonomous differential equation, \(\frac{dy}{dt} = f(y)\), we can use the graph of the rate function to understand the behavior of the solution function. When the rate function is positive \(f(y) \gt 0\) (above the axis), the solution \(y(t)\) will be an increasing function. When the rate function is negative \(f(y) \lt 0\) (below the axis), the solution \(y(t)\) will be a decreasing function. Equilibrium solutions correspond to values of \(y\) where \(f(y)=0\). We can summarize this behavior by drawing a number line with arrows pointing in the direction of increasing or decreasing between equilibrium values. Such a number line is called a phase line.