# Preface

This web page is dedicated to autonomous systems of differential equation that are generated by linear systems of ordinary differential equations (ODEs for short). A linear first order ordinary differential equation can be used as a mathematical model for a variety of phenomena, either physical or non-physical. Examples of such phenomena include the following: heat flow problems (thermodynamics), simple electrical circuits (electrical engineering), force problems (mechanics) , rate of bacterial growth (biological science) , rate of decomposition of radioactive material (atomic physics) , crystallization rate of a chemical compound (chemistry), and rate of population growth (statistics).

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Introduction to Linear Algebra with *Mathematica*

## Glossary

Much of the progress of nineteenth century science was due to the realisation and systematic exploitation of the fact that many of the laws of nature were linear. That is to say that, given two solutions

*y*

_{1}and

*y*

_{2}of a problem, they could be combined together linearly to give further solutions

*c*

_{1}

*y*

_{1}+

*c*

_{2}

*y*

_{2}for some constants

*c*

_{1},

*c*

_{2}. The idea can be traced at least as far back as Newton who used it in his explanation as to why certain ports (like Southampton) have four and not two tides per day. These ideas of linearity exerted a profound influence even outside the circle of professional mathematicians and physicists. Alexander Bell’s initial experiments were intended to find a method for simultaneously conveying several Morse code messages along the same telegraph wire by using different frequencies.

# The topics to be covered in this chapter:

# 2.1: Variable coefficient systems of ODEs

This section presents basic properties of solutions to the linear systems of differential equations with variable coefficients.

# 2.2: Constant coefficient systems of ODEs

This section presents basic properties of solutions to the linear systems of differential equations with variable coefficients.

# 2.3: Planar Phase Portraits

Two dimensional autonomous systems of differential equations can be fully visualized with with phase portraits that represent typical trajectories on the plane.

# 2.4: Euler systems of equations

# 2.5: Fundamental matrices

# 2.6: Reduction to a single equation

# 2.7: Method of undetermined coefficients

# 2.8: Variation of parameters

# 2.9: Laplace transform

# 2.10: Second order ODEs

# 2.11: Spring-mass systems

# 2.12: Electric circuits

# 2.13: Applications