6: Linear Systems

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"Do not worry too much about your difficulties in mathematics, I can assure you that mine are still greater." - Albert Einstein ( \(1879-1955\) )

6.1: Linear Systems
IN SECTION 3.5 WE SAW THAT the numerical solution of second order equations, or higher, can be cast into systems of first order equations. Such systems are typically coupled in the sense that the solution of at least one of the equations in the system depends on knowing one of the other solutions in the system. In many physical systems this coupling takes place naturally. We will introduce a simple model in this section to illustrate the coupling of simple oscillators.
6.2: Applications
IN THIS SECTION WE WILL DESCRIBE SOME SIMPLE APPLICATIONS leading to systems of differential equations which can be solved using the methods in this chapter. These systems are left for homework problems and the as the start of further explorations for student projects.
6.3: Matrix Formulation
We have investigated several linear systems in the plane and in the next chapter we will use some of these ideas to investigate nonlinear systems. We need a deeper insight into the solutions of planar systems. So, in this section we will recast the first order linear systems into matrix form. This will lead to a better understanding of first order systems and allow for extensions to higher dimensions and the solution of nonhomogeneous equations later in this chapter.
6.4: Eigenvalue Problems
We seek nontrivial solutions to the eigenvalue problem
6.5: Solving Constant Coefficient Systems in 2D
Before proceeding to examples, we first indicate the types of solutions that could result from the solution of a homogeneous, constant coefficient system of first order differential equations.
6.6: Examples of the Matrix Method
Here we will give some examples for typical systems for the three cases mentioned in the last section.
6.7: Theory of Homogeneous Constant Coefficient Systems
There is a general theory for solving homogeneous, constant coefficient systems of first order differential equations.
6.8: Nonhomogeneous Systems
Before leaving the theory of systems of linear, constant coefficient systems, we will discuss nonhomogeneous systems.
6.9: Problems

Thumbnail: Example spiral source vector field. (CC BY-SA 4.0; Jiří Lebl via Source)