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Contents

1 Setting the Scene 1
1.1 What Is a Differential Equation? ............... 1
1.1.1 Concepts ........................ 2
1.2 The Solution and Its Properties ................ 4
1.2.1 An Ordinary Differential Equation .......... 4
1.3 A Numerical Method ...................... 6
1.4 Cauchy Problems ........................ 10
1.4.1 First-Order Homogeneous Equations ......... 10
1.4.2 First-Order Nonhomogeneous Equations ....... 13
1.4.3 The Wave Equation .................. 15
1.4.4 The Heat Equation ................... 18
1.5 Exercises ............................ 20
1.6 Projects ............................. 28

2 Two-Point Boundary Value Problems 39
2.1 Poisson’s Equation in One Dimension ............ 40
2.1.1 Green’s Function .................... 42
2.1.2 Smoothness of the Solution .............. 43
2.1.3 A Maximum Principle ................. 44
2.2 A Finite Difference Approximation .............. 45
2.2.1 Taylor Series ...................... 46
2.2.2 A System of Algebraic Equations ........... 47
2.2.3 Gaussian Elimination for Tridiagonal Linear Systems 50
2.2.4 Diagonal Dominant Matrices ............. 53