Future Books

Advanced Topics in Calculus: Differential Equations

John Hubbard and Barbara Burke Hubbard

The tentative publication date for this book is December 2025.

Incomplete table of contents (draft)

Chapter 1 Differential equations as models

1.0 Introduction
1.1 Initial value differential equations
1.2 Differential equations and the exponential
1.3 Applications to ecology: sharks and sardines
1.4 The pendulum and one-dimensional dynamics
1.5 An application to astronomy
1.6 Boundary-value problems: soap bubbles
1.7 Review exercises for Chapter 1

Chapter 2 Qualitative methods

2.0 Introduction
2.1 First-order equations: fences and funnels
2.2 Fences for equations of higher dimension
2.3 Review exercises for Chapter 2

Chapter 3 Analytic methods: first-order equations

3.0 Introduction
3.1 Separation of variables
3.2 Exact equations and differential forms
3.3 The Frobenius theorem
3.4 Review exercises for Chapter 3

Chapter 4 Existence and uniqueness of solutions

4.1 The fundamental inequality
4.2 Uniqueness of solutions
4.3 Euler's method
4.4 Existence of solutions
4.5 Applications to qualitative methods
4.6 Flows of differential equations
4.7 A surprising case of Euler's method
4.8 Review exercises for Chapter 4




Chapter 5 Analytic methods: linear equations with constant coefficients

5.0 Introduction
5.1 What makes linear equations linear?
5.2 Higher-order linear equations with constant coefficients
5.3 Beyond the cookbook: behavior of solutions
5.4 Eigenvectors, eigenvalues, and decoupling
5.5 Partial differential equations and eigenvalues
5.6 Exponentials of non-diagonalizable matrices
5.7 Nonhomogeneous equations: solving with undetermined coefficients
5.8 Nonhomogeneous equations: solving with variation of parameters
5.9 Review exercises for Chapter 5

Chapter 6 Numerical methods

6.1 A survey of numerical methods
6.2 Some experimental results
6.3 Runge-Kutta methods
6.4 The discrete fundamental inequality
6.5 Round-off errors
6.6 How should one use numerical methods?
6.7 Implicit methods
6.8 Symplectic integrators
6.9 Review exercises for Chapter 6

Chapter 7 Second-order linear equations with nonconstant coefficients

7.0 Introduction
7.1 Existence and uniquness of solutions
7.2 The two Prüfer transforms
7.3 Sturm-Liouville theory
7.4 Regular singular points
7.5 Bessel functions
7.6 Review exercises for Chapter 7

Chapter 8 Planar vector fields

8.0 Introduction
8.1 Some planar examples
8.2 Sinks and sources
8.3 Saddles and separatrices in the plane
8.4 The Poincaré-Bendixon theorem
8.5 Periodic solutions and limit cycles
8.6 Structurally stable planar differential equations
8.7 Review exercises for Chapter 8

Chapter 9 Planar systems: bifurcations

9.1 Introduction
9.2 Saddle nodes
9.3 Andronov-Hopf bifurcations
9.4 Saddle connections
9.5 Semi-stable limit cycles
9.6 Some codimension 2 bifurcations
9.7 Grand examples
9.8 Review exercises for Chapter 9

Chapter 10 Unstable manifolds in higher dimensions

10.1 Change of variables in dynamical systems
10.2 Poincaré maps in higher dimensions
10.3 Formal linearization
10.4 Sinks are sinks and sources are sources
10.5 Unstable manifolds and N-prepared maps


Chapter 11 Smale's horseshoe

Chapter 10 The homoclinic tangle