3.1: Introduction

Most of your studies of differential equations to date have been the study linear differential equations and common methods for solving them. However, the real world is very nonlinear. So, why study linear equations? Because they are more readily solved. As you may recall, we can use the property of linear superposition of solutions of linear differential equations to obtain general solutions. We will see that we can sometimes approximate the solutions of nonlinear systems with linear systems in small regions of phase space.

In general, nonlinear equations cannot be solved obtaining general solutions. However, we can often investigate the behavior of the solutions without actually being able to find simple expressions in terms of elementary functions. When we want to follow the evolution of these solutions, we resort to numerically solving our differential equations. Such numerical methods need to be executed with care and there are many techniques that can be used. We will not go into these techniques in this course. However, we can make use of computer algebra systems, or computer programs, already developed for obtaining such solutions.

Nonlinear problems occur naturally. We will see problems from many of the same fields we explored in Section 2.9. One example is that of population dynamics. Typically, we have a certain population, \(y(t)\), and the differential equation governing the growth behavior of this population is developed in a manner similar to that used previously for mixing problems. We note that the rate of change of the population is given by the Rate In minus the Rate Out. The Rate In is given by the number of the species born per unit time. The Rate Out is given by the number that die per unit time.

A simple population model can be obtained if one assumes that these rates are linear in the population. Thus, we assume that the Rate In = by and the Rate Out = my. Here we have denoted the birth rate as \(b\) and the mortality rate as \(m\), . This gives the rate of change of population as

\[\dfrac{dy}{dt} = by - my \label{3.1} \]

Generally, these rates could depend upon time. In the case that they are both constant rates, we can define \(k = b − m\) and we obtain the familiar exponential model:

\[\dfrac{dy}{dt} = ky \nonumber \]

This is easily solved and one obtains exponential growth (\(k > 0\)) or decay (\(k < 0\)). This model has been named after Malthus, a clergyman who used this model to warn of the impending doom of the human race if its reproductive practices continued.

However, when populations get large enough, there is competition for resources, such as space and food, which can lead to a higher mortality rate. Thus, the mortality rate may be a function of the population size, \(m=m(y)\). The simplest model would be a linear dependence, \(m=\tilde{m}+c y\). Then, the previous exponential model takes the form

\[\dfrac{dy}{dt} = ky - cy^2 \label{3.2} \]

This is known as the logistic model of population growth. Typically, \(c\) is small and the added nonlinear term does not really kick in until the population gets large enough.

While one can solve this particular equation, it is instructive to study the qualitative behavior of the solutions without actually writing down the explicit solutions. Such methods are useful for more difficult nonlinear equations. We will investigate some simple first order equations in the next section. In the following section we present the analytic solution for completeness.

We will resume our studies of systems of equations and various applications throughout the rest of this chapter. We will see that we can get quite a bit of information about the behavior of solutions by using some of our earlier methods for linear systems.