When mathematical modelling is used to describe physical, biological or chemical phenomena, one of the most common results is either a differential equation or a system of differential equations, together with appropriate boundary and initial conditions. These differential equations may be ordinary or partial, and finding and interpreting their solution is at the heart of applied mathematics. A thorough introduction to differential equations is therefore a necessary part of the education of any applied mathematician, and this book is aimed at building up skills in this area. For similar reasons, the book should also be of use to mathematically-inclined physicists and engineers.
Although the importance of studying differential equations is not generally in question, exactly how the theory of differential equations should be taught, and what aspects should be emphasized, is more controversial. In our experience, textbooks on differential equations usually fall into one of two categories. Firstly, there is the type of textbook that emphasizes the importance of abstract mathematical results, proving each of its theorems with full mathematical rigour. Such textbooks are usually aimed at graduate students, and are inappropriate for the average undergraduate. Secondly, there is the type of textbook that shows the student how to construct solutions of differential equations, with particular emphasis on algorithmic methods. These textbooks often tackle only linear equations, and have no pretension to mathematical rigour. However, they are usually well-stocked with interesting examples, and often include sections on numerical solution methods.