10: Linear Systems of Differential Equations — Read

From LibreTexts, under CC BY-NC-SA 3.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

IN THIS CHAPTER we consider systems of differential equations involving more than one unknown function. Such systems arise in many physical applications.SECTION 10.1 presents examples of physical situations that lead to systems of differential equations. SECTION 10.2 discusses linear systems of differential equations. SECTION 10.3 deals with the basic theory of homogeneous linear systems. SECTIONS 10.4, 10.5, AND 10.6 present the theory of constant coefficient homogeneous systems. SECTION 10.7 presents the method of variation of parameters for nonhomogeneous linear systems.

10.1: Introduction to Systems of Differential Equations
Many physical situations are modeled by systems of n differential equations in n unknown functions, where n≥2 . This section presents examples of physical situations that lead to systems of differential equations.
- 10.1.1: Introduction to Systems of Differential Equations (Exercises)
10.2: Linear Systems of Differential Equations
A first order system of differential equations are introduced.
- 10.2.1: Linear Systems of Differential Equations (Exercises)
10.3: Basic Theory of Homogeneous Linear Systems
In this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with the theory of linear homogeneous scalar equations.
- 10.3.1: Basic Theory of Homogeneous Linear Systems (Exercises)
10.4: Constant Coefficient Homogeneous Systems I
We’ll now begin our study of the homogeneous system y′=Ay, where A is an n×n constant matrix. . In this section we assume that all the eigenvalues of A are real and that A has a set of n linearly independent eigenvectors. In the next two sections we consider the cases where some of the eigenvalues of A are complex, or where A does not have n linearly independent eigenvectors.
- 10.4.1: Constant Coefficient Homogeneous Systems I (Exercises)
10.5: Constant Coefficient Homogeneous Systems II
In this section we consider the case where A has n real eigenvalues, but does not have n linearly independent eigenvectors. It is shown in linear algebra that this occurs if and only if A has at least one eigenvalue of multiplicity r>1 such that the associated eigenspace has dimension less than r . In this case A is said to be defective. We will restrict our attention to some commonly occurring special cases.
- 10.5.1: Constant Coefficient Homogeneous Systems II (Exercises)
10.6: Constant Coefficient Homogeneous Systems III
We now consider the system y′=Ay , where A has a complex eigenvalue λ=α+iβ with β≠0 . We continue to assume that A has real entries, so the characteristic polynomial of A has real coefficients. This implies that λ=α−iβ is also an eigenvalue of A .
- 10.6.1: Constant Coefficient Homogeneous Systems III (Exercises)
10.7: Variation of Parameters for Nonhomogeneous Linear Systems
We now discuss an extension of the method of variation of parameters to linear nonhomogeneous systems. This method will produce a particular solution of a nonhomogenous system y′=A(t)y+f(t) provided that we know a fundamental matrix for the complementary system.
- 10.7.1: Variation of Parameters for Nonhomogeneous Linear Systems (Exercises)

This page titled 10: Linear Systems of Differential Equations was last modified on Tue, 07 Jan 2020 03:44:21 GMT and is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

Contents

Text