To this point we’ve only looked at solving single
differential equations. However, many
“real life” situations are governed by a system of differential equations. Consider the population problems that we
looked at back in the modeling section of the first
order differential equations chapter. In
these problems we looked only at a population of one species, yet the problem
also contained some information about predators of the species. We assumed that any predation would be
constant in these cases. However, in
most cases the level of predation would also be dependent upon the population
of the predator. So, to be more realistic
we should also have a second differential equation that would give the population
of the predators. Also note that the
population of the predator would be, in some way, dependent upon the population
of the prey as well. In other words, we
would need to know something about one population to find the other
population. So to find the population of
either the prey or the predator we would need to solve a system of at least two
differential equations.
The next topic of discussion is then how to solve systems of
differential equations. However, before
doing this we will first need to do a quick review of Linear Algebra. Much of what we will be doing in this chapter
will be dependent upon topics from linear algebra. This review is not intended to completely
teach you the subject of linear algebra, as that is a topic for a complete
class. The quick review is intended to
get you familiar enough with some of the basic topics that you will be able to
do the work required once we get around to solving systems of differential
equations.
Here is a brief listing of the topics covered in this
chapter.
Review : Systems of Equations The traditional starting point for a linear
algebra class. We will use linear
algebra techniques to solve a system of equations.
Review : Matrices and Vectors A brief introduction to matrices and
vectors. We will look at arithmetic
involving matrices and vectors, inverse of a matrix, determinant of a matrix,
linearly independent vectors and systems of equations revisited.
Review : Eigenvalues and Eigenvectors
Finding the eigenvalues and eigenvectors of a
matrix. This topic will be key to
solving systems of differential equations.
Systems of Differential Equations Here we will look at some of the basics of
systems of differential equations.
Solutions to Systems We will take a look at what is involved in
solving a system of differential equations.
Phase Plane A brief introduction to the phase plane and
phase portraits.
Real Eigenvalues Solving systems of differential equations with
real eigenvalues.
Complex Eigenvalues Solving systems of differential equations with
complex eigenvalues.
Repeated Eigenvalues Solving systems of differential equations with
repeated eigenvalues.
Nonhomogeneous Systems Solving nonhomogeneous systems of differential
equations using undetermined coefficients and variation of parameters.
Laplace Transforms A very brief look at how Laplace transforms
can be used to solve a system of differential equations.
Modeling In this section we’ll take a quick look at
some extensions of some of the modeling we did in previous chapters that lead
to systems of equations.