An Introduction to Ordinary Differential Equations
This book grew out of lecture notes prepared by the author for the first year Warwick differential equation course. It covers a wide scale of elementary methods of obtaining explicit solutions to ordinary first order differential equations and to linear equations and systems of two linear equations with constant coefficients and special cases of nonconstant coefficients (Euler equations) with remarks on power series solutions. One chapter is devoted to numerical solutions (Euler's method) and difference equations. Special attention is given to the dynamical system approach, starting with the qualitative behaviour for one equation (stability, bifurcation) and phase portraits of linear systems and moving on to nonlinear systems modelling pendulum motion and predator-prey systems, the Poincaré - Bendixson theorem and remarks on chaos. More complicated dynamics are shown on the Lorentz system. An overview of needed background material is given in appendices A, B and C, dealing with real and complex numbers, elements of linear algebra and derivatives and Taylor expansions. Needed elements of integration are contained in the main text. The presentation is vivid and informal, careful to minimize requirements on the reader's previous knowledge. New ideas and general concepts are well documented on worked examples. The text is richly illustrated (with figures generated mostly by Matlab). Individual work by the student is encouraged by exercises at the end of each section (including exercises suggested for computer aided graphical investigation) and a list of books recommended for further reading. The book will be useful to anybody wanting to teach or learn elements of ordinary differential equations in the beginnings of their mathematical studies.