Preface ......................................................... v
1 Introduction ................................................. 1
1.1 Differential equations .................................. 1
1.2 Second order differential equations ..................... 3
1.3 Differential equations for functions on  p .............. 9
1.4 Multidimensional differential equations ................ 15
1.5 Overview ............................................... 19
1.6 Exercises .............................................. 20
2 Expansions with orthogonal polynomials ...................... 23
2.1 Introduction ........................................... 23
2.2 Laguerre's polynomials ................................. 25
2.3 Hermite's polynomials .................................. 30
2.4 Legendre's polynomials ................................. 38
2.5 Generalizations ........................................ 42
2.6 Bilinear functions ..................................... 44
2.7 Hermite's polynomials in 2 ............................ 47
2.8 Legendre's polynomials in 2 ........................... 50
2.9 Exercises .............................................. 55
3 Differential and integral calculus .......................... 57
3.1 Differentiability of functions ......................... 57
3.2 Maximum and minimum of functions ....................... 59
3.3 Euler-Lagrange conditions .............................. 63
3.4 Integral calculus ...................................... 67
3.5 Partial derivatives of elliptic functions .............. 71
3.6 Applications ........................................... 80
3.7 Exercises .............................................. 87
4 Linear differential equations ............................... 89
4.1 First order differential equations in + ............... 89
4.2 Second order linear differential equations in + ....... 95
4.3 Sturm-Liouville second order differential equations ... 108
4.4 Applications .......................................... 112
4.5 Nonlinear differential equations ...................... 113
4.6 Differential equations of higher orders ............... 119
4.7 Exercises ............................................. 122
5 Linear differential equations in P ........................ 125
5.1 Introduction .......................................... 125
5.2 Laplace's differential equation ....................... 130
5.3 Potential equations ................................... 132
5.4 Heat conduction equations ............................. 137
5.5 Wave differential equations ........................... 146
5.6 Elasticity equations .................................. 154
5.7 Exercises ............................................. 156
6 Partial differential equations ............................. 157
6.1 Partial differential equations in 2 .................. 157
6.2 First order linear partial differential equations ..... 159
6.3 Second order linear partial differential equations .... 170
6.4 Multidimensional differential equations ............... 171
6.5 Lotka-Volterra equations .............................. 177
6.6 Birth-and-death differential equations ................ 181
6.7 Poincare-Lorenz differential system ................... 184
6.8 Exercises ............................................. 187
7 Special functions .......................................... 189
7.1 Eulerian functions .................................... 189
7.2 Airy function ......................................... 196
7.3 Bessel's function ..................................... 198
7.4 Boyd's function ....................................... 199
7.5 Hermite's function .................................... 199
7.6 Laguerre's functions .................................. 201
7.7 Hydrogen atom equation and others ..................... 202
7.8 Exercises ............................................. 203
8 Solutions .................................................. 205
8.1 Integral and differential calculus .................... 205
8.2 Orthogonal polynomials ................................ 211
8.3 Calculus and optimization ............................. 214
8.4 Linear differential equations ......................... 220
8.5 Linear differential equations in P ................... 226
8.6 Partial differential equations ........................ 228
8.7 Special functions ..................................... 231
8.8 Programs .............................................. 235
Bibliography .................................................. 241
Index ......................................................... 243
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