Originally published in 1820, this is an early work by the renowned mathematician and inventor Charles Babbage (1791–1871). The text was written to provide mathematical students with an accessible introduction to functional equations, an area that had been previously absent from elementary mathematical literature. A short bibliography is also contained. This book will be of value to anyone with an interest in Babbage and the history of mathematics.

Part I Ordinary Differential Equations

1 First-Order Ordinary Differential Equations 1

1.1 Basics 1

1.2 Special Equations 8

2 Higher Order Ordinary Differential Equations 17

2.1 Basics 17

2.2 Method of Undetermined Coefficients 21

2.3 Method of Variation of Parameters 25

2.4 Series Method and Bessel Functions 29

3 Secial Functions 37

3.1 Gamma and Beta Functions 37

3.2 Gauss Hypergeometric Functions 43

3.3 Orthogonal Polynomials 47

3.4 Weierstrass's Elliptic Functions 54

3.5 Jacobian Elliptic Functions 61

Part II Partial Differential Equations

4 First-Order or Linear Equations 67

4.1 Method of Characteristics 68

4.2 Characteristic Strip and Exact Equations 71

4.3 Polynomial Solutions of Flag Equations 74

4.4 Use of Fourier Expansion I 93

4.5 Use of Fourier Expansion II 100

4.6 Calogero-Sutherland Model 117

4.7 Maxwell Equations 126

4.8 Dirac Equation and Acoustic System 134

5 Nonlinear Scalar Equations 141

5.1 Korteweg and de Vries Equation 142

5.2 Kadomtsev and Petviashvili Equation 149

5.3 Equation of Transonic Gas Flows 155

5.4 Short-Wave Equation 161

5.5 Khokhlov and Zabolotskaya Equation 168

5.6 Equation of Geopotential Forecast 172

6 Nonlinear Schrödinger and Davey-Stewartson Equations 179

6.1 Nonlinear Schrödinger Equation 179

6.2 Coupled Schrödinger Equations 187

6.3 Davey and Stewartson Equations 201

7 Dynamic Convection in a Sea 213

7.1 Equations and Symmetries 213

7.2 Moving-Line Approach 216

7.3 Cylindrical Product Approach 219

7.4 Dimensional Reduction 223

8 Boussinesq Equations in Geophysics 231

8.1 Two-Dimensional Equations 231

8.2 Three-Dimensional Equations and Symmetry 247

8.3 Asymmetric Approach I 249

8.4Asymmetric Approach II 255

8.5 Asymmetric Approach III 61

9 Navier-Stokes Equations 269

9.1 Background and Symmetry 269

9.2 Asymmetric Approaches 273

9.3 Moving-Frame Approach I 285

9.4Moving-Frame Approach II 296

10 Classical Boundary Layer Problems 317

10.1 Two-Dimensional Problem 317

10.2 Three-Dimensional Problem: General 326

10.3 Uniform Exponential Approaches 332

10.4 Distinct Exponential Approaches 344

10.5 Trigonometric and Hyperbolic Approaches 350

10.6 Rational Approaches 362

References 385

Index 393

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