A dynamic process is a phenomenon that changes its attributes over time. Most dynamic processes are modeled by using ordinary differential equations (ODEs), which were invented to model the rate of change of a phenomenon. There are two ways to attack ODEs: symbolically or numerically. For most real-life problems, there are no symbolic solutions, and thus we concentrate on numerical methods.

9.1 Background We start with a simple example. One of the oldest models to study population growth goes back to T. Malthus around 1800. Malthus was interested in predicting human population size over many years. He invented a simple model, which states that population grows at a rate proportional to the existing population.1 If p(t) denotes population at a given time t, then p′(t) is the rate of change at that time-for example, if p′(t) = 0, then population does not change at time t. Thus