So you want to learn about differential equations? Excellent choice. Differential equations are not only of central importance to science, they can also be quite stimulating and fun (that’s right). In the broadest sense, a differential equation is any equation that connects a function with one or more of its derivatives. What makes these kinds of equations particularly important?
Remember that a derivative expresses the rate of change of a quantity. So the differential equation basically establishes a link between the rate of change of said quantity and its current value. Such a link is very common in nature. Consider population growth. It is obvious that the rate of change will depend on the current size of the population. The more animals there are, the more births (and deaths) we can expect and hence the faster the size of the population will change.
A commonly used model for population growth is the exponential model. It is based on the assumption that the rate of change is proportional to the current size of the population. Let’s put this into mathematical form. We will denote the size of the population by N (measured in number of animals) and the first derivative with respect to time by dN/dt (measured in number of animals per unit time). Note that other symbols often used for the first derivative are N’ and Ṅ. We will however stick to the so-called Leibniz notation dN/dt as it will prove to be quite instructive when dealing with separable differential equations. That said, let’s go back to the exponential model.
With N being the size of the population and dN/dt the corresponding rate of change, our assumption of proportionality between the two looks like this:
with r being a constant. We can interpret r as the growth rate. If r > 0, then the population will grow, if r < 0, it will shrink. This model has proven to be successful for relatively small animal populations. However, there’s one big flaw: there is no limiting value. According to the model, the population would just keep on growing and growing until it consumes the entire universe. Obviously and luckily, bacteria in Petri dish don’t behave this way. For a more accurate model, we need to take into account the limits of the environment.
The differential equation that forms the basis of the logistic model, called Verhulst equation in honor of the Belgian mathematician Pierre François Verhulst, does just that. Just like the differential equation for exponential growth, it relates the current size N of the population to its rate of change dN/dt, but also takes into account the finite capacity K of the environment:
Take a careful look at the equation. Even without any calculations a differential equation can tell a vivid story. Suppose for example that the population is very small. In this case N is much smaller than K, so the ratio N/K is close to zero. This means that we are back to the exponential model. Hence, the logistic model contains the exponential model as a special case. Great! The other extreme is N = K, that is, when the size of the population reaches the capacity. In this case the ratio N/K is one and the rate of change dN/dt becomes zero, which is exactly what we were expecting. No more growth at the capacity.
Definition and Equation of Motion
Now that you have seen two examples of differential equations, let’s generalize the whole thing. For starters, note that we can rewrite the two equations as such:
Denoting the dependent variable with x (instead of N) and higher order derivatives with dnx/dtn (with n = 2 resulting in the second derivative, n = 3 in the third derivative, and so on), the general form of a differential equation looks like this:
Wow, that looks horrible! But don’t worry. We just stated in the broadest way possible that a differential equation is any equation that connects a function x(t) with one or more its derivatives dx/dt, d2x/dt2, and so on. The above differential equation is said to have the order n. Up to now, we’ve only been dealing with first order differential equations.
The following equation is an example of a second order differential equation that you’ll frequently come across in physics. Its solution x(t) describes the position or angle over time of an oscillating object (spring, pendulum).
with c being a constant. Second order differential equations often arise naturally from Newton’s equation of motion. This law, which even the most ruthless crook will never be able to break, states that the object’s mass m times the acceleration a experienced by it is equal to the applied net force F:
The force can be a function of the object’s location x (spring), the velocity v = dx/dt (air resistance), the acceleration a = d2x/dt2 (Bremsstrahlung) and time t (motor):
Hence the equation of motion becomes:
A second order differential equation which leads to the object’s position over time x(t) given the forces involved in shaping its motion. It might not look pretty to some (it does to me), but there’s no doubt that it is extremely powerful and useful.
To demonstrate what equilibrium points are and how to compute them, let’s take the logistic model a step further. In the absence of predators, we can assume the fish in a certain lake to grow according to Verhulst’s equation. The presence of fishermen obviously changes the dynamics of the population. Every time a fisherman goes out, he will remove some of the fish from the population. It is safe to assume that the success of the fisherman depends on the current size of the population: the more fish there are, the more he will be able to catch. We can set up a modified version of Verhulst’s equation to describe the situation mathematically:
with a constant c > 0 that depends on the total number of fishermen, the frequency and duration of their fishing trips, the size of their nets, and so on. Solving this differential equation is quite difficult. However, what we can do with relative ease is finding equilibrium points.
Remember that dN/dt describes the rate of change of the population. Hence, by setting dN/dt = 0, we can find out if and when the population reaches a constant size. Let’s do this for the above equation.
This leads to two solutions:
The first equilibrium point is quite boring. Once the population reaches zero, it will remain there. You don’t need to do math to see that. However, the second equilibrium point is much more interesting. It tells us how to calculate the size of the population in the long run from the constants. We can also see that a stable population is only possible if c < r.
Note that not all equilibrium points that we find during such an analysis are actually stable (in the sense that the system will return to the equilibrium point after a small disturbance). The easiest way to find out whether an equilibrium point is stable or not is to plot the rate of change, in this case dN/dt, over the dependent variable, in this case N. If the curve goes from positive to negative values at the equilibrium point, the equilibrium is stable, otherwise it is unstable.
(This was an excerpt from my e-book “Math Shorts – Introduction to Differential Equations”)