A first tour through examples - Chapter 1

The simplest example

The simplest differential equation is

$$\dot{x}=ax$$

where $x\in\mathbb{R}$ and $a$ is a real-valued parameter. Despite its simplicity, this equation allows us to introduce several important concepts for the rest of this ebook. We can interpret it as a naive model for the growth of bacterial populations.

Bacteria.
Consider a container filled with a nutritive solution and some bacteria. As time progresses, the bacteria reproduce (by division) and die. Let $b$ (for birth) be the rate at which the bacteria reproduce and $d$ (for death) be the rate at which they die. Then, the net growth of the population is done at rate $b-d$. This means that if there are $x$ bacteria in the container, then the rate at which the number of bacteria is increasing is $(b-d)x$, that is,

$$\dot{x}=ax$$

where $a=b-d$. What does exactly mean this equation? It means that $x=x(t)$ is an unknown real-valued function of a real variable $t$, interpreted as time, and $\dot{x}$ is its derivative. In this ebook we will use most of the time the notation $\dot{x}$ for $\frac{\text{d}x}{\text{d}t}$.
For each value of the parameter $a$ we have a differential equation. The equation tells us that for every value of $t$ the relationship $\dot{x}(t)=ax(t)$ is true. It can be solved explicitly from calculus (by separation of variables): the function $x(t)=c e^{at}$ is a solution where $c$ is an arbitrary constant of integration. Moreover, there are no other solutions.


To see this, let $y(t)$ be any solution and compute the time derivative of $y(t)\, e^{-rt}$:

$$ \frac{\text{d}}{\text{d}t} \big(y(t)\, e^{-at}\big) = \dot{y}(t)\,e^{-at}+y(t)(-a\,e^{-at}) = ay(t)\, e^{-at}-ay(t)\, e^{-at}=0.$$

Hence $y(t)\, e^{-at}$ is a constant $c$, so $y(t)=ce^{at}$. This proves our assertion.


We have therefore found all possible solutions of this differential equation. To determine completely one particular solution, we have to find the constant $c$. This is possible if we specify the value $x_0$ of a solution at a single time $t_0$. Indeed, if we require that $x(t_0)=x_0$ then we must have $ce^{at_0}=x_0$, so that $c=x_0\, e^{a t_0}$. Having determined $c$, we therefore have a unique solution satisfying the initial condition $x(t_0)=x_0$. With the interpretation we have in mind, we take $x_0>0$ since a quantity of bacteria is a positive number. When $x_0=0$, $x(t)\equiv 0$; this constant solution is called a fixed point for the equation.

For simplicity we can take $t_0=0$; then $c=x_0$. There is no loss of generality in doing that, for if $x(t)$ is a solution with $x(0)=x_0$, then the function $x^*(t)=x(t-t_0)$ is a solution with $x^*(t_0)=x_0$.
Therefore, given $x(0)=x_0$, the differential equation $\dot{x}=ax$ has a unique solution $x(t)=x_0 e^{at}$. Its behavior depends on the sign of $a$:

  • If $a>0$, $\lim_{t\to+\infty} x(t)=+\infty$ for all $x_0>0$;
  • If $a=0$, $x(t)=x_0$ for all $t\geq 0$;
  • If $a<0$, $\lim_{t\to+\infty} x(t)=0$ for all $x_0>0$.

If births prevail over deaths, that is, if $b>d$, then $a>0$ and the population increases exponentially to $+\infty$. If deaths prevail over births then $a<0$ and the population goes extinct exponentially fast. When births and deaths exactly compensate each other, the population size remain constant. According to the equation, the same duration of time is always required to double the population if $a>0$, or to reduce it by one-half if $a<0$, regardless of its size.
Forgetting about modeling aspects, the equation $\dot{x}=ax$ can of course be studied for any real-valued initial condition $x_0$, not necessarily positive. To complete the above list of behaviors, we have $\lim_{t\to+\infty} x(t)=-\infty$ when $a>0$ and $x_0<0$. When $a<0$, the fate of the solution is independent of the sign of $x_0$.

Needless to say that the equation $\dot{x}=ax$ is encountered in a large number of other problems, for instance to describe radioactive decay ($a<0$).


Robustness and bifurcation.
The equation $\dot{x}=ax$ is robust in a certain sense if $a\neq 0$. More precisely, if $a$ is replaced by another constant $a’$ whose sign is the same as $a$, then the qualitative behavior of the solutions does not change. But if $a=0$, the slightest change in $a$ leads to a dramatic change in the behavior of solutions. We say that we have a bifurcation at $a=0$ in the one-parameter family of equations $\dot{x}=ax$. We will see many examples of (more interesting) bifurcations later on.