Introduction to Differential Equations

Robert B. Israel
Department of Mathematics
University of British Columbia

What is a differential equation?

A differential equation is an equation involving one or more derivatives of an unknown function. Here are some examples:

1.

$$\frac{dy}{dt} = k y$$

This is one you've already seen in Math 101: the law of exponential growth. The unknown function is y (which is to be considered as a function of t). We call t the independent variable and y the dependent variable; k is a constant. The solutions of this differential equation are

$$y = A {\rm e}^{k t}$$

where A is an arbitrary constant. That is, for any constant A, the function $y(t) = A \exp(k t)$ satisfies the equation. As we will see, the general solution of a differential equation always involves at least one arbitrary constant. You can determine its value if you give some additional information, such as the value of y at t=0 (an initial condition). This differential equation is a first-order equation because it only involves the first derivative of y and no higher derivatives.

2.

$$\frac{dx}{dt} = \cos t$$

This is another one you've seen, although you didn't call it a differential equation then. You can solve it by integrating:

$$x = \int \cos t \, dt = \sin t + C$$

Again we have the arbitrary constant C. This was also a first-order equation.

3.

$$\frac{d^2 y}{d t^2} = - g$$

The equation of a falling body. Since it involves the second derivative of the dependent variable y, it is a second-order equation. This one can be solved by two integrations:

$$y = - \frac{g}{2} t^2 + A t + B$$

where A and B are arbitrary constants.

4.

$$\frac{dx}{dt} = \cos(x t)$$

A first order equation, but one you probably haven't seen before. I don't know any solutions to this one (I don't think anyone else does either). Nevertheless, I can tell you a lot about the solutions.

I don't want to give you the impression that the subject of differential equations is mainly concerned with solving them explicitly, i.e. finding algebraic expressions for the solutions. It's actually quite rare for that to be possible. There are other aspects we'll see that are at least as important:

• Numerical solutions, usually obtained by computer.
• Qualitative methods for studying the behaviour of solutions, in particular graphical methods.

5.

$$\frac{d^3 y}{d t^3} + \cos(t) \frac{dy}{dt} + {\rm e}^{t} y = t$$

This is a third-order equation. It is linear: each of its terms can have an arbitrary function of t, times 1 or y or a derivative of y, but no other dependence on y. All our previous examples except (4) were also linear, (4) was nonlinear. Linear equations tend to be much easier to deal with than nonlinear ones, and have important properties that we will see. In fact, much of our understanding of nonlinear equations is based on approximating them with linear equations.

6.

$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$

This is a partial differential equation ( PDE ) : it has two independent variables x and y. We won't study it in this course. The differential equations we study, with only one independent variable, are called ordinary differential equations (ODE's). You'll meet some PDE's, including this one, if you take Math 316.

7.

$$\begin{eqnarray*}x' &=& - x + y \\ y' &=& - x z + x - y \\ z' &=& x y - z \end{eqnarray*}$$

Here ' means derivative with respect to t. This is a system of ODE's. We will eventually study some systems as well as individual ODE's.