Coordinated Differential Equations

Coordinated Differential Equations is adapted from Thomas W. Judson’s Open Educational Resource (OER) The Ordinary Differential Equations Project, originally published under a GNU Free Documentation License in 2018. This text was modified from the original to better fit the curriculum of MATH 221–Differential Equations at the University of Nebraska - Lincoln.

While the spirit of the original text is preserved in this version, there are some structural differences. Perhaps most significantly, the treatment of second-order linear equations now immediately follows the introductory material in Chapter 1. In Judson’s original, the second chapter is devoted to linear systems of equations; second-order equations are covered later in the text using techniques from linear systems. (In particular, the homogeneous second-order equation \(ax'' + bx' + cx = 0\) is solved by introducing an extra variable \(u = x'\text{,}\) which transforms the original equation into a two-by-two linear system.) In Coordinated Differential Equations, second-order linear equations are solved solely using the characteristic polynomial, with all linear algebra material deferred to later chapters.

In order to give a richer picture of solutions to second-order linear equations, a new section in Chapter 1 is devoted entirely to complex numbers and their role in differential equations. This section includes basic information on complex arithmetic, as well as deeper material on Euler’s formula (which is derived by solving the complex initial value problem \(z' = iz\text{,}\) \(z(0) = 1\)) and various trigonometric identities. The chapter on the Laplace Transform also occupies a more prominent place in Coordinated Differential Equations than in the original, immediately following Chapter 2 on second-order linear equations.

While some exercises have been slighlty modified, and a few new ones added, the vast majority of the exercises in Coordinated Differential Equations are identical to those in The Ordinary Differential Equations Project. One caveat in this regard is with respect to exercises for which a computer algebra system may be useful or necessary: the original text contains many "built-in" SageMath modules, while the modified text does not. This is due to the fact that Sage is not commonly used in UNL mathematics courses, MATLAB being the more typical choice. As such, most references to Sage were removed during the developent of Coordinated Differential Equations. (Of course, interested students are still encouraged to explore other tools, including Sage, on their own!).

Judson’s original preface is included below for completeness. The present authors are heavily indebted to his work in developing the original text, which gives a clear, concise, and highly engaging introduction to ordinary differential equations.