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Differential Equations (Notes) / Systems of DE's   [Notes]
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 Systems of Differential Equations

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.

Table Of Laplace Transforms Previous Section   Next Section Review : Systems of Equations
Laplace Transforms Previous Chapter   Next Chapter Series Solutions to DE's

Differential Equations (Notes) / Systems of DE's    [Notes]

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