Differential Equations: An Introduction to Modern Methods and Applications – J. Brannan and W. Boyce

Ah, differential equations. Few topics in mathematics have been shown to be quite as useful or polarizing than diffEQ. Engineers of all types likely love to hate them, but rely on them heavily. Biologists and chemists need them as well. Really, anyone touching science or engineering will encounter the differential equation. Therefore, it’s important to select a good textbook, for there is certainly no shortage out there. I used this text for my undergrad differential equations course; it’s meant as an introduction. I consider the book fundamental in its approach, much like a good calculus book. One must build the skills in recognizing and solving some of the more common types of differential equations (linear, first-order, separable, etc) while being given a hint of the heavier math behind them. Heavier discussions on existence of solutions and stability can be reserved for the next level course, and this book does an admirable job of teaching the reader with many different examples, good word problems and discussion, illustration of application, and a plethora of exercises to practice. Each chapter is concluded with a summary. I highly recommend the use of this book for a one-semester undergraduate introduction to differential equations, following multivariate calculus and a little bit of linear algebra. The book is good for self-study if one is pretty fresh on his calculus, or needs to review diffEQ. I’ll certainly teach out of it if I ever get the chance1

-Rachel Traylor, Ph.D.



single and multivariable calculus, some matrix algebra

Topics Covered

  • First order differential equations
    • separable equations
    • integrating factors
    • numerical methods
  • systems of two first order equations
    • systems of linear algebraic equations
    • homogenous linear systems with constant coefficients
    • complex/repeated eigenvalues
  • second order linear equations
    • characteristic equations
    • applications in physics
    • variation of parameters
  • Laplace transform
    • convolutions and applications
  • systems of first order linear equations
    • fundamental matrices 
    • matrix exponentials
    • non homogenous systems
  • nonlinear differential equations and stability
    • predator-prey equations
    • Lorenz equations


  1. Backstory, I taught 5 years at two different universities, and never did get assigned diffEQ…always wanted to teach it though.


Difficulty 3
Good for teaching? 5
Proof Quality NA
Quality of Exercises 5
Self-Study 3
Readability 4