Introduction to Systems of ODEs

Systems of ODEs are important in various fields of science, such as the study of electricity and population biology. Like single ODEs, systems of ODEs can classified as linear or nonlinear.

A system of linear first-order ODEs can be represented in the form

Here is a vector of unknown functions, is the matrix of the coefficients of the unknown functions, and is a vector representing the inhomogeneous part of the system.

In the two-dimensional case, the system can be written more concretely as

If all the entries of the matrix are constants, then the system is said to be linear with constant coefficients. If is the zero vector, then the system is said to be homogeneous.

The important global features of the solutions to linear systems can be clarified by considering homogeneous systems of ODEs with constant coefficients.