This chapter introduces the eigenvalue-eigenvector method for finding solutions of autonomous homogeneous linear systems of differential equations, namely, systems of the form \(\displaystyle \frac {dx}{dt} = ax + by, \quad \frac {dy}{dt} = cx + dy, \qquad \hbox{(1)} \) where a, b, c, d are constants. Prior knowledge of the matrix algebra that is used in this section to find solutions of (1) is not assumed in the text since many students enrolled in an introductory differential equations course have little to no knowledge of matrix algebra, especially if the course is taken immediately after the first two semesters of calculus. Accordingly, this textbook is written so that it can be read without previous exposure to matrices: any terminology and results from matrix algebra that are needed to understand this chapter have been woven into the text rather than being relegated to an appendix. Sections 12.1 through 12.4 contain foundational information for understanding the rest of the chapter. Section 12.1 introduces terminology related to (1). An example in Sect. 12.2 illustrates that linear systems as simple as (1) can serve as mathematical models of certain types of physical systems, such as modeling the amount of sugar at any given time in each of several interconnected tanks composing a multi-tank system if pure water is pumped into the system while sugar water is being pumped out at the same rate. Section 12.3 defines vectors, matrices, and the operations involving them, such as matrix addition and multiplication, after which it is shown how to write (1) more compactly as the vector differential equation \(\displaystyle \mathbf {x}'= A\mathbf {x}, \;\: \text{where} \;\: \mathbf {x} = \begin {bmatrix} x \\ y \end {bmatrix} \; \text{and} \;\: A = \begin {bmatrix} a & b \\ c & d \end {bmatrix}. \qquad \hbox{(2)} \) Section 12.4 introduces terms, such as velocity vector field, direction field, phase point, and trajectory, which aid in describing the state of an imaginary particle whose motion if governed by (2) for given values of a, b, c, and d. Section 12.5 begins with an example of an uncoupled linear system and its solutions as a means to assist in devising a method to solve any system (1) when it is rewritten as (2). This leads to the definitions of eigenvalues and eigenvectors of a matrix followed by more definitions, such as determinant, inverse, and characteristic equation of a matrix. Topics presented in this section related to (2) and its solutions are the Superposition and Autonomous Principles, linear independence of functions, Wronskian of solutions, Abel’s formula, just to name a few. The section culminates with theorems (and their proofs) regarding the solutions of (2) and an ample number of examples explaining how to find a general solution of a given equation taking into account whether the eigenvalues of the coefficient matrix A are real (distinct or repeated) or complex numbers. The chapter concludes with Sect. 12.6, where linear systems (1) are used to model the temperatures of one- and two-room buildings.

Planar Autonomous Homogeneous Linear Systems

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发表日期：

2026年

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Leigh C. Becker

摘要

This chapter introduces the eigenvalue-eigenvector method for finding solutions of autonomous homogeneous linear systems of differential equations, namely, systems of the form \(\displaystyle \frac {dx}{dt} = ax + by, \quad \frac {dy}{dt} = cx + dy, \qquad \hbox{(1)} \) where a, b, c, d are constants. Prior knowledge of the matrix algebra that is used in this section to find solutions of (1) is not assumed in the text since many students enrolled in an introductory differential equations course have little to no knowledge of matrix algebra, especially if the course is taken immediately after the first two semesters of calculus. Accordingly, this textbook is written so that it can be read without previous exposure to matrices: any terminology and results from matrix algebra that are needed to understand this chapter have been woven into the text rather than being relegated to an appendix. Sections 12.1 through 12.4 contain foundational information for understanding the rest of the chapter. Section 12.1 introduces terminology related to (1). An example in Sect. 12.2 illustrates that linear systems as simple as (1) can serve as mathematical models of certain types of physical systems, such as modeling the amount of sugar at any given time in each of several interconnected tanks composing a multi-tank system if pure water is pumped into the system while sugar water is being pumped out at the same rate. Section 12.3 defines vectors, matrices, and the operations involving them, such as matrix addition and multiplication, after which it is shown how to write (1) more compactly as the vector differential equation \(\displaystyle \mathbf {x}'= A\mathbf {x}, \;\: \text{where} \;\: \mathbf {x} = \begin {bmatrix} x \\ y \end {bmatrix} \; \text{and} \;\: A = \begin {bmatrix} a & b \\ c & d \end {bmatrix}. \qquad \hbox{(2)} \) Section 12.4 introduces terms, such as velocity vector field, direction field, phase point, and trajectory, which aid in describing the state of an imaginary particle whose motion if governed by (2) for given values of a, b, c, and d. Section 12.5 begins with an example of an uncoupled linear system and its solutions as a means to assist in devising a method to solve any system (1) when it is rewritten as (2). This leads to the definitions of eigenvalues and eigenvectors of a matrix followed by more definitions, such as determinant, inverse, and characteristic equation of a matrix. Topics presented in this section related to (2) and its solutions are the Superposition and Autonomous Principles, linear independence of functions, Wronskian of solutions, Abel’s formula, just to name a few. The section culminates with theorems (and their proofs) regarding the solutions of (2) and an ample number of examples explaining how to find a general solution of a given equation taking into account whether the eigenvalues of the coefficient matrix A are real (distinct or repeated) or complex numbers. The chapter concludes with Sect. 12.6, where linear systems (1) are used to model the temperatures of one- and two-room buildings.