An Introduction to Ordinary Differential Equations

Description

This refreshing, introductory textbook covers both standard techniques for solving ordinary differential equations, as well as introducing students to qualitative methods such as phase-plane analysis. The presentation is concise, informal yet rigorous; it can be used either for 1-term or 1-semester courses. Topics such as Euler's method, difference equations, the dynamics of the logistic map, and the Lorenz equations, demonstrate the vitality of the subject, and provide pointers to further study. The author also encourages a graphical approach to the equations and their solutions, and to that end the book is profusely illustrated. The files to produce the figures using MATLAB are all provided in an accompanying website. Numerous worked examples provide motivation for and illustration of key ideas and show how to make the transition from theory to practice. Exercises are also provided to test and extend understanding: solutions for these are available for teachers. Cover Half-title Title Copyright Dedication Contents Preface Introduction Part I First order differential equations 1 Radioactive decay and carbon dating 1.1 Radioactive decay 1.2 Radiocarbon dating Exercises 2 Integration variables 3 Classification of differential equations 3.1 Ordinary and partial differential equations 3.2 The order of a differential equation 3.3 Linear and nonlinear 3.4 Different types of solution Exercises 4 Graphical representation of solutions using MATLAB Exercises 5 ‘Trivial’ differential equations 5.1 The Fundamental Theorem of Calculus 5.2 General solutions and initial conditions 5.3 Velocity, acceleration and Newton’s second law of motion 5.4 An equation that we cannot solve explicitly Exercises 6 Existence and uniqueness of solutions 6.1 The case for an abstract result 6.2 The existence and uniqueness theorem 6.3 Maximal interval of existence 6.4 The Clay Mathematics Institute’s $1 000 000 question Exercises 7 Scalar autonomous ODEs 7.1 The qualitative approach 7.2 Stability, instability and bifurcation 7.3 Analytic conditions for stability and instability 7.4 Structural stability and bifurcations 7.5 Some examples 7.5.1 A population model 7.5.2 Terminal velocity 7.5.3 What have we lost? 7.6 The pitchfork bifurcation 7.7 Dynamical systems Exercises 8 Separable equations 8.1 The solution ‘recipe’ 8.2 The linear equation… 8.2.1 Exponential decay and exponential growth 8.3 Malthus’ population model 8.4 Justifying the method 8.5 A more realistic population model 8.6 Further examples 8.6.1 Partial fractions again 8.6.2 Two competing species Exercises 9 First order linear equations and the integrating factor 9.1 Constant coefficients 9.2 Integrating factors 9.3 Examples 9.4 Newton’s law of cooling 9.4.1 Estimating the time of death 9.4.2 The temperature in an unheated building 9.4.3 Combining two oscillating terms 9.4.4 Back to our example Exercises 10 Two ‘tricks’ for nonlinear equations 10.1 Exact equations 10.1.1 Integrating factors 10.2 Substitution methods 10.2.1 Homogeneous equations 10.2.2 Bernoulli equations Exercises Part II Second order linear equations with constant coefficients 11 Second order linear equations: general theory 11.1 Existence and uniqueness 11.2 Linearity 11.3 Linearly independent solutions 11.3.1 Linear independence of functions 11.3.2 Two linearly independent solutions are necessary and sufficient 11.4 The Wronskian 11.5 Linear algebra Exercises 12 Homogeneous second order linear equations with constant coefficients 12.1 Two distinct real roots 12.2 A repeated real root 12.3 No real roots Exercises 13 Oscillations 13.1 The spring 13.2 The simple pendulum 13.3 Damped oscillations Over-damping Critical damping Under-damping Exercises 14 Inhomogeneous second order linear equations 14.1 Complementary function and particular integral 14.2 When f(t) is a polynomial 14.3 When f(t) is an exponential 14.4 When f(t) is a sine or cosine 14.5 Rule of thumb 14.6 More complicated functions f(t) Exercises 15 Resonance 15.1 Periodic forcing 15.1.1 No resonance: bounded response 15.1.2 ‘Ideal’ resonance: unbounded response 15.2 Pseudo resonance in physical systems Exercises 16 Higher order linear equations with constant coefficients 16.1 Complementary function and particular integral 16.2 The general theory for nth order equations Exercises Part III Linear second order equations with variable coefficients 17 Reduction of order Exercises 18 The variation of constants formula Exercises 19 Cauchy–Euler equations 19.1 Two real roots 19.2 A repeated root 19.3 Complex roots Exercises 20 Series solutions of second order linear equations 20.1 Power series 20.2 Ordinary points 20.3 Regular singular points 20.4 Bessel’s equation Exercises Part IV Numerical methods and difference equations 21 Euler’s method 21.1 Euler’s method 21.2 An example 21.3 MATLAB implementation of Euler’s method 21.4 Convergence of Euler’s method Exercises 22 Difference equations 22.1 First order difference equations 22.2 Second order difference equations: complementary function and particular solution 22.3 The homogeneous equation 22.3.1 Distinct real roots 22.3.2 Repeated roots 22.3.3 Complex roots 22.4 Particular solutions 22.4.1 Right-hand side fn is a polynomial in n 22.4.2 Right-hand side… Exercises 23 Nonlinear first order difference equations 23.1 Fixed points and stability 23.2 Cobweb diagrams 23.3 Periodic orbits 23.4 Euler’s method for autonomous equations 23.4.1 An example Exercises 24 The logistic map 24.1 Fixed points and their stability 24.2 Periodic orbits 24.3 The period-doubling cascade 24.4 The bifurcation diagram and more periodic orbits 24.5 Chaos 24.6 Analysis of… Exercises Part V Coupled linear equations 25 Vector first order equations and higher order equations 25.1 Existence and uniqueness for second order equations Exercises 26 Explicit solutions of coupled linear systems Exercises 27 The matrix approach to linear equations: eigenvalues and eigenvectors 27.1 Rewriting the equation in matrix form 27.2 Eigenvalues and eigenvectors 27.3 Eigenvalues and eigenvectors with MATLAB Exercises 28 Distinct real eigenvalues 28.1 The explicit solution 28.2 Changing coordinates 28.3 Phase diagrams for uncoupled equations 28.4 Phase diagrams for coupled equations 28.5 Stable and unstable manifolds Exercises 29 More phase portraits: complex eigenvalues 29.1 The explicit solution 29.2 Changing coordinates and the phase portrait 29.3 The phase portrait for the original equation Exercises 30 Yet more phase portraits: a repeated real eigenvalue 30.1 A is a multiple of the identity: stars 30.2 A is not a multiple of the identity: improper nodes Exercises 31 Summary of phase portraits for linear equations 31.1 Jordan canonical form 31.1.1 Representation of vectors in different coordinate systems 31.1.2 Linear transformations of the plane and 2 × 2 matrices 31.1.3 Similar matrices and the Jordan canonical form Exercises Part VI Coupled nonlinear equations 32 Coupled nonlinear equations 32.1 Some comments on phase portraits 32.2 Competition of species 32.3 Direction fields 32.4 Analytical method for phase portraits 32.4.1 Step 1: find the stationary points 32.4.2 Step 2: linearise near the stationary points 32.4.3 The Hartman–Grobman Theorem 32.4.4 Step 3: find the stability type of each stationary point 32.4.5 Step 4: ‘join the dots’ Exercises 33 Ecological models 33.1 Competing species 33.1.1 Weak competition 33.1.2 Coexistence 33.2 Predator-prey models I 33.3 Predator-prey models II Exercises 34 Newtonian dynamics 34.1 One-dimensional conservative systems 34.2 A bead on a wire 34.3 Dissipative systems Exercises 35 The ‘real’ pendulum 35.1 The undamped pendulum 35.2 The damped pendulum 35.3 Alternative phase space Exercises 36 Periodic orbits 36.1 Dulac’s criterion 36.2 The Poincaré–Bendixson Theorem Exercises 37 The Lorenz equations 38 What next? 38.1 Partial differential equations and boundary value problems 38.2 Dynamical systems and chaos Exercises Appendix A: Real and complex numbers Real numbers Complex numbers Appendix B: Matrices, eigenvalues, and eigenvectors Basic matrix algebra Matrices and vectors Multiplication of vectors by matrices Solution of simultaneous equations Eigenvalues and eigenvectors Linear independence of eigenvectors The special case of symmetric matrices Appendix C: Derivatives and partial derivatives Functions of one variable: ordinary derivatives Definition and properties of the derivative Taylor expansions Power series Turning points Functions of two variables: partial derivatives Partial derivatives and their properties Taylor expansions Critical points Index