Linear Systems in Differential Equations

Understanding Linear Systems

A linear system of differential equations is a set of equations involving multiple functions and their derivatives, all of which are linear in nature.
Linear systems appear frequently in fields like physics, engineering, and economics, often when modelling interconnected systems.
Linear systems can be written in matrix form, which often simplifies the process of finding solutions.

Components of Linear Systems

Linear systems are typically ordered by order, which refers to the highest derivative present in the system.
They are also structured by the number of unknowns, which refers to the number of functions being solved for.
Most systems encountered at this level will be first order and involve two unknowns.

Homogeneous and Non-Homogeneous Linear Systems

A linear system is said to be homogeneous if all of its equations equal zero. These systems often have simpler solutions, and are a good starting point for understanding linear systems.
In contrast, a non-homogeneous linear system is one in which not all the equations equal zero. The solutions to these systems are generally more complex.

Approaches to Solving Linear Systems

Often, the first step to solving a linear system is to write it in matrix form.
The eigenvalue method is a commonly used approach when dealing with homogeneous linear systems. This involves calculating the eigenvalues and eigenvectors of the matrix form of the system - these will form the basis for the system’s general solution.
For non-homogeneous systems, one approach is to find a particular solution that fits the non-homogeneous term, and combine this with the general solution from the corresponding homogeneous system.
Alternatively, the method of undetermined coefficients or variation of parameters may be used to find solutions in these cases.

Linear Independence, Dependence, and the Wronskian

The solutions to linear systems are often expressed in terms of basis functions, which can take on many forms but are often exponential functions or sine/cosine waves. The basis functions need to be linearly independent for the solution to be valid.
Independence can be checked using the Wronskian, a determinant that is constructed from the solutions and their derivatives. If the Wronskian is non-zero for all values of the independent variable (often time or position), then the functions are linearly independent.

Don’t forget, the study of linear systems of differential equations is an important part of mathematics, particularly in the engineering fields. Understanding the different components, solution techniques, and verification methods for these systems lays the foundation for many advanced topics.