An overview of ordinary differential equations

A differential equation is an equation that expresses the relationship between an unknown function, say f(t), and the derivatives of f(t). A differential equation is called an 'ordinary' differential equation (ODE) if the function that we are seeking depends on a single independent variable, as does the function f(t). This is in contrast to 'partial' differential equations (PDEs), in which the solution function depends on two or more independent variables, as in f(x, t). ODE's often describe dynamic systems, that is, systems that change with time. However, one should keep in mind that ODE's are not limited to describing only time-varying systems.

Some ODE's have analytic solutions, and these solutions are very helpful in understanding the dynamic behavior of a system. Whole math courses are taught on the subject of ODE's and ways to analytically solve them. Some engineering courses are devoted mostly to solving ODE's related to a particular type of process, chemical engineering for instance.

"Solving" an ODE consists of answering the following question: what function can you construct such that it has the relationship to its derivatives that the ODE expresses? There is no single procedure for finding this function. A great deal of intuition and experience is necessary to analytically solve ODE's. This is why software like Maple is so powerful for solving ODE's.

Many times the ODE does not have an analytic solution, and a numerical solution must be sought. We will look at both analytical and numerical solutions to first order ODE's. An ODE is first order if only the first derivative of the function is included. If an ODE is higher than first order, we will see how it can be manipulated to give a system of first order ODEs.

First order ODE's relate the first derivative of a function (time rate of change) with the function itself. Perhaps the simplest meaningful example is:

The function that is the solution to this equation is

Exponential growth or decay is governed by this simple ODE. Substitute y(t) into the ODE to check that this is the solution. In engineering, we can often describe a process using relations between functions and their derivatives. Here are some common situations in which ODEs are often used.

• Radioactive decay
• Chemical processes
• Controls of almost anything
• Population dynamics
• Diffusion of almost anything
• Transient electrical circuits

For a reference, let's look at a comparison between ODE's and linear systems.