What is a differential equation?

Ordinary differential equations: Introduction

What is a differential equation?

Equations as you know them You are probably already acquainted with equations whose solutions are numbers. For example, the solutions of the quadratic equation \(x^2=2\) are \(x=\sqrt{2}\) and \(x=-\sqrt{2}\).

You are probably also familiar with equations whose solutions are pairs of numbers. For example, the solution of the equation \(x^2+y^2=1\) is the set of all pairs of numbers \((x,y)\) that are the coordinates of points on the circle with centre \((0,0)\) and radius 1. So there is an infinite number of solutions. Given a number \(x\) between \(-1\) and \(1\) there are only two solutions, namely \(y=\sqrt{1-x^2}\) and \(y=-\sqrt{1-x^2}\). The positive solution \(y=\sqrt{1-x^2}\) can be viewed as a functional relationship between \(y\) and \(x\). In other words, the equation \(x^2+y^2=1\) has a function as a solution. It basically addresses the question which function \(f(x)\) has the property that \(f(x)^2+x^2=1\) for all \(x\).

A differential equation A differential equation is an equation whose solution is a function (or sometimes a set of functions). But in this case, in addition to one or more unknown functions, also one or more derivatives of these functions are present in the equation. The unknown in a differential equation is not a number but a function (of place, time, or both).

A differential equation for a function of a single variable is called an ordinary differential equation, abbreviated ODE.
A differential equation for a function of two or more variables is called a partial differential equation, abbreviated PDE, because the derivatives present in the equation are so-called partial derivatives.

Examples of ODEs Examples of ODEs are:

Exponential growth or decay \[y'(t) = r\cdot y(t)\] with a constant \(r\neq 0\).
Limited exponential growth: \[y'(t) = r\cdot \bigl(a- y(t)\big)\] with nonnegative constants \(r\) and \(a\).
Logistic growth: \[y'(t) = r\cdot y(t)\cdot\left(1-\frac{y(t)}{a}\right)\] with constants \(r, a\neq 0\).

Examples of PDEs Examples of PDEs are:

Diffusion equation for the propagation of an action potential along an axon: \[\frac{\partial v(x,t)}{\partial t} = a \frac{\partial^2 v(x,t)}{\partial x^2} -b\] where \(a\) and \(b\) are nonnegative constants.
Fisher equation for a space-time description of population growth: \[\frac{\partial y(x,t)}{\partial t} = D \frac{\partial^2 y(x,t)}{\partial x^2} + r\, y(x,t)\,\left(1-\frac{y(x,t)}{K}\right)\] with constants \(D, r, K\neq 0\).

Partial derivatives A partial derivative is a concept for functions of several variables. We briefly describe what it means. For a function \(f(x,t)\) that depends on location \(x\) and time \(t\) you can keep one of the variables, say \(t\), constant. Then you get a function in one variable, namely with \(x\) as an independent variable. When this function in one variable is a neat function, you can determine its derivative: this derivative is called the partial derivative of \(f(x,t)\) with respect to \(x\) of order 1, which is denoted as \(\dfrac{\partial f(x,t)}{\partial t}(x,t)\). You can view the partial derivative as a function of two variables and again compute partial derivatives. Then you get second-order partial derivatives such as: \[\frac{\partial^2 f(x,t)}{\partial x^2}=\frac{\partial}{\partial x}\left(\frac{\partial f(x,t)}{\partial x}\right)\quad\text{and}\quad \frac{\partial^2 f(x,t)}{\partial x \partial t}=\frac{\partial}{\partial x}\left(\frac{\partial f(x,t)}{\partial t}\right)\]

The function \[f(x,y)=2x^2+3xy+4y^2\] has the following partial derivatives of order 1 and 2: \[\begin{aligned} \frac{\partial f}{\partial x} &= 4x+3y \\ \\ \frac{\partial f}{\partial y} &= 3x+8y \\ \\ \frac{\partial^2 f}{\partial x^2} &= 4\\ \\ \frac{\partial^2 f}{\partial y^2} &= 8\\ \\ \frac{\partial^2 f}{\partial x\partial y} &= \frac{\partial^2 f}{\partial y\partial x}=3\end{aligned}\]

In this chapter, we only study functions of one variable, mostly time \(t\) so that these ODEs are so-called dynamical systems. Partial differential equations are not covered.

Example 1 A simple example of a differential equation is \[y'(t)=0\tiny.\] The unknown function is \(y(t)\), with the property that its derivative is equal to zero. A solution of this differential equation is \(y(t)=1\), because the derivative is indeed 0. But there are other solutions: \(y(t)=0\) and \(y(t)=1\!\tfrac{1}{2}\) are also solutions. In fact, any constant function is a solution of the given differential equation and we write it as: \[y(t)=c\] where \(c\) is any constant. This initially unknown function can therefore also be defined as the general solution of the given differential equation. However, there are infinitely many solutions as \(c\) can take any real value. To select a single solution one needs an extra condition: for example, the value of the function at a certain point in time.

Example 2 A more complicated example of a differential equation is \[y'(t)\cdot y(t)=t\tiny.\] The unknown function is \(y(t)\), with the property that the product of the function with its derivative is equal to the identity function. The general solution of this differential equation is \(y(t)=\sqrt{t^2+c}\) or \(y(t)=-\sqrt{t^2+c}\,\). To understand the correctness of this solution and how it can be found, it helps to rewrite the left-hand side of the given ODE as \(\tfrac{1}{2}\cdot\left(y(t)^2\right)' \). So the derivative of the square of the unknown function equals \(t\mapsto 2t\), which is in itself the derivative of the quadratic function \(t\mapsto t^2\). This is why \(y(t)^2 = t^2+c\), for some constant \(c\).