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.