Math Insight

Solving single autonomous differential equations using graphical methods


Video introduction

A graphical approach to solving an autonomous differential equation.

More information about video.


One can understand an autonomous differential equation of the form \begin{align} \diff{x}{t} &= f(x)\\ x(t_0) &= x_0\notag \end{align} by using a purely graphical approach. We can determine the essential behavior of the solution $x(t)$ without doing any analytic calculations. A graph of the function $f(x)$ will tell us all we need to know to estimate what the solution $x(t)$ will do for any initial condition $x_0$.

Since the derivative $\diff{x}{t}$ is the rate of change of $x(t)$, a glance at the graph of $f(x)$ will tell us where $x(t)$ is increasing or decreasing and how fast it is changing. The state variable $x(t)$ moves to larger values when $f(x)$ is positive, and it moves to smaller values when $f(x)$ is negative. The velocity of $x(t)$ drops to zero when $f(x)$ reaches zero. The points where $f(x)=0$ are the equilibria where $x(t)$ does not move.

An example

For example, we'll look at the differential equation \begin{align*} \diff{x}{t} = x^2-4. \end{align*} We can determine the dynamics of the solution $x(t)$ by looking at the graph of $f(x)=x^2-4$.

Graph of $f(x)=x^2-4$

From the graph we see that $f(x) < 0$ for $x \in (-2,2)$. The rate of change $\diff{x}{t}$ must be negative for $-2 \lt x(t) \lt 2$, so the solution decreases in that range. If the initial condition $x_0$ were in the range $x_0 \in (-2,2)$, then the solution would start out decreasing. The speed of this negative change would be greatest when $x(t)=0$, and then the trajectory would slow down as it got closer to $x(t)=-2$.

The function $f(x)$ is zero at $x=2$ and $x=-2$. Therefore, these two values of $x(t)$ are equilbria. If the initial condition $x_0$ were $x_0=-2$, then the solution would be a constant $x(t)=-2$ for all time. Similarly, if the initial condition were $x_0=2$, then the solution would be the constant $x(t)=2$ for all time. For the case mentioned above, with initial condition $x_0 \in (-2,2)$, the trajectory would get closer and closer to $-2$. It could never cross $x=-2$ because we know the velocity is zero at that point.

One solution to an initial condition just below 2 is graphed below. In the graph, $x(t)$ is plotted as a function of $t$. In this plot, the $x$-axis has moved to the vertical axis, and the $t$-axis is the horizontal axis. Notice that $x(t)$ decreases the whole time, decreases most quickly around $x=0$, then slows down, getting closer and closer to $x=-2$ as the time $t$ increases.

One solution to $dx/dt=x^2-4$.

What changes if the initial condition is below $x=-2$. In that case, the function $f(x)$ is positive, so the rate of change $\diff{x}{t}$ is positive. If the initial condition $x_0 \lt -2$, then the solution $x(t)$ starts out increasing. If $x_0$ is much smaller than $-2$, then $x(t)$ increases very quickly, but its velocity slows down as $x(t)$ approaches the equilibrium $x=-2$. It continues to increase the whole time, as it can't cross the equilibrium, but its velocity goes to zero as it gets closer to the equilibrium.

The graph of $f(x)$ is symmetric across the vertical axis. So, it might seem that something similar will happen for initial conditions above the upper equilibrium $x=2$. However, the behavior is entirely different. Just as in the case for $x_0 \lt -2$, for initial condition $x_0 \gt 2$, the trajectory starts off with a positive rate of change $\diff{x}{t}$. In this case, though, as $x(t)$ increases, its velocity increases even more. The trajectory quickly blows up to very large values.

In the following applet, you can explore the behavior of solutions to $\diff{x}{t} = x^2-4$ for many different initial conditions. You can observe how initial conditions below $x=2$ lead to solutions $x(t)$ that converge toward $x=-2$. You can also see how quickly solutions with initial conditions $x_0 \gt 2$ blow up. This applet will also help you solidify the relationship between the graph of $f(x)$ (where $x$ is the horizontal axis) and the graph of the trajectories versus time (where $x$ is the vertical axis).

Exploring autonomous differential equations.

More information about applet.

The second applet lets you see the simultaneous behavior of six solutions with different initial conditions. Before you hit play or move the $t$ slider, see if you can predict what the solutions will look like.

Exploring autonomous differential equations, multiple trajectories.

More information about applet.