Cover......Page 1
Handbook of First Order Differential Equations......Page 3
Contents......Page 5
Preface......Page 15
Authors......Page 16
Annotation......Page 17
Some Notation and Remarks......Page 18
Part I. Linear Equations With Two Independent Variables......Page 19
1. Equations Containing One Derivative......Page 20
2.1.1. Solution Method......Page 21
2.1.3. Examples......Page 22
2.2.1. Coefficients of Equations Are Linear in x and y......Page 23
2.2.2. Coefficients of Equations Are Quadratic in x and y......Page 25
2.2.3. Coefficients of Equations Contain Integer Powers of x and y......Page 28
2.2.4. Coefficients of Equations Contain Fractional Powers......Page 29
2.2.5. Coefficients of Equations Contain Arbitrary Powers of x and y......Page 30
2.3.1. Coefficients of Equations Contain Exponential Functions......Page 40
2.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions......Page 41
2.4.2. Coefficients of Equations Contain Hyperbolic Cosine......Page 46
2.4.3. Coefficients of Equations Contain Hyperbolic Tangent......Page 47
2.4.4. Coefficients of Equations Contain Hyperbolic Cotangent......Page 48
2.4.5. Coefficients of Equations Contain Different Hyperbolic Functions......Page 49
2.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions......Page 50
2.6.1. Coefficients of Equations Contain Sine......Page 53
2.6.2. Coefficients of Equations Contain Cosine......Page 55
2.6.3. Coefficients of Equations Contain Tangent......Page 56
2.6.4. Coefficients of Equations Contain Cotangent......Page 58
2.6.5. Coefficients of Equations Contain Different Trigonometric Functions......Page 59
2.7.1. Coefficients of Equations Contain Arcsine......Page 61
2.7.2. Coefficients of Equations Contain Arccosine......Page 62
2.7.3. Coefficients of Equations Contain Arctangent......Page 64
2.7.4. Coefficients of Equations Contain Arccotangent......Page 65
2.8.1. Equations Contain Arbitrary and Power-Law Functions......Page 67
2.8.2. Equations Contain Arbitrary and Exponential Functions......Page 69
2.8.3. Equations Contain Arbitrary and Hyperbolic Functions......Page 70
2.8.5. Equations Contain Arbitrary and Trigonometric Functions......Page 71
2.8.6. Equations Contain Arbitrary Functions and Their Derivatives......Page 72
2.9.1. Equations Contain Arbitrary Functions of x and Arbitrary Functions of y......Page 74
2.9.2. Equations Contain One Arbitrary Function of Complicated Argument......Page 75
2.9.3. Equations Contain Several Arbitrary Functions......Page 77
3.1.1. Solution Methods......Page 81
3.1.2. Cauchy Problem......Page 82
3.1.3. Examples......Page 83
3.2.1. Coefficients of Equations Are Linear in x and y......Page 84
3.2.2. Coefficients of Equations Are Quadratic in x and y......Page 85
3.2.4. Coefficients of Equations Contain Arbitrary Powers of x and y......Page 86
3.3.1. Coefficients of Equations Contain Exponential Functions......Page 88
3.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions......Page 89
3.4.1. Coefficients of Equations Contain Hyperbolic Sine......Page 90
3.4.2. Coefficients of Equations Contain Hyperbolic Cosine......Page 91
3.4.4. Coefficients of Equations Contain Hyperbolic Cotangent......Page 92
3.5.1. Coefficients of Equations Contain Logarithmic Functions......Page 93
3.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions......Page 94
3.6.2. Coefficients of Equations Contain Cosine......Page 95
3.6.4. Coefficients of Equations Contain Cotangent......Page 96
3.6.5. Coefficients of Equations Contain Different Trigonometric Functions......Page 97
3.7.1. Coefficients of Equations Contain Arcsine......Page 98
3.7.3. Coefficients of Equations Contain Arctangent......Page 99
3.7.4. Coefficients of Equations Contain Arccotangent......Page 100
3.8.1. Coefficients of Equations Contain Arbitrary Functions of x......Page 101
3.8.2. Equations Contain Arbitrary Functions of x and Arbitrary Functions of y......Page 103
3.8.3. Equations Contain Arbitrary Functions of Complicated Arguments......Page 104
3.8.4. Equations Contain Arbitrary Functions of Two Variables......Page 105
4.1.1. Solution Methods......Page 107
4.1.2. Examples......Page 108
4.2.1. Coefficients of Equations Are Linear in x and y......Page 109
4.2.2. Coefficients of Equations Are Quadratic in x and y......Page 110
4.2.3. Coefficients of Equations Contain Other Power-Law Functions......Page 111
4.2.4. Coefficients of Equations Contain Arbitrary Powers of x and y......Page 112
4.3.1. Coefficients of Equations Contain Exponential Functions......Page 113
4.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions......Page 114
4.4.1. Coefficients of Equations Contain Hyperbolic Sine......Page 115
4.4.3. Coefficients of Equations Contain Hyperbolic Tangent......Page 116
4.4.4. Coefficients of Equations Contain Hyperbolic Cotangent......Page 117
4.5.1. Coefficients of Equations Contain Logarithmic Functions......Page 118
4.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions......Page 119
4.6.2. Coefficients of Equations Contain Cosine......Page 120
4.6.3. Coefficients of Equations Contain Tangent......Page 121
4.6.5. Coefficients of Equations Contain Different Trigonometric Functions......Page 122
4.7.1. Coefficients of Equations Contain Arcsine......Page 123
4.7.3. Coefficients of Equations Contain Arctangent......Page 124
4.8.1. Coefficients of Equations Contain Arbitrary Functions of x......Page 125
4.8.2. Equations Contain Arbitrary Functions of x and Arbitrary Functions of y......Page 128
4.8.3. Equations Contain Arbitrary Functions of Complicated Arguments......Page 129
4.8.4. Equations Contain Arbitrary Functions of Two Variables......Page 130
5.1.1. Solution Methods......Page 131
5.1.2. Examples......Page 132
5.2.1. Coefficients of Equations Are Linear in x and y......Page 133
5.2.2. Coefficients of Equations Are Quadratic in x and y......Page 134
5.2.3. Coefficients of Equations Contain Square Roots......Page 135
5.2.4. Coefficients of Equations Contain Arbitrary Powers of x and y......Page 136
5.3.1. Coefficients 0 f Equations Contain Exponential Functions......Page 138
5.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions......Page 139
5.4.2. Coefficients of Equations Contain Hyperbolic Cosine......Page 141
5.4.4. Coefficients of Equations Contain Hyperbolic Cotangent......Page 142
5.5.1. Coefficients of Equations Contain Logarithmic Functions......Page 143
5.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions......Page 144
5.6.1. Coefficients of Equations Contain Sine......Page 145
5.6.3. Coefficients of Equations Contain Tangent......Page 146
5.6.4. Coefficients of Equations Contain Cotangent......Page 147
5.6.5. Coefficients of Equations Contain Different Trigonometric Functions......Page 148
5.7.2. Coefficients of Equations Contain Arccosine......Page 149
5.7.4. Coefficients of Equations Contain Arccotangent......Page 150
5.8.1. Coefficients of Equations Contain Arbitrary Functions of x......Page 151
5.8.2. Equations Contain Arbitrary Functions of x and Arbitrary Functions of y......Page 153
5.8.3. Equations Contain Arbitrary Functions of Two Variables......Page 154
Part II. Linear Equations With Three or More Independent Variables......Page 155
6.1.1. Solution Methods......Page 156
6.1.3. Examples......Page 157
6.2.1. Coefficients of Equations Are Linear in x, y, and z......Page 158
6.2.2. Coefficients of Equations Are Quadratic in x. y, and z......Page 162
6.2.3. Coefficients of Equations Contain Other Powers of x, y, and z......Page 165
6.2.4. Coefficients of Equations Contain Arbitrary Powers of x, y, and z......Page 166
6.3.1. Coefficients of Equations Contain Exponential Functions......Page 169
6.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions......Page 170
6.4.1. Coefficients of Equations Contain Hyperbolic Sine......Page 172
6.4.2. Coefficients of Equations Contain Hyperbolic Cosine......Page 173
6.4.4. Coefficients of Equations Contain Hyperbolic Cotangent......Page 174
6.4.5. Coefficients of Equations Contain Different Hyperbolic Functions......Page 175
6.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions......Page 176
6.6.2. Coefficients of Equations Contain Cosine......Page 177
6.6.4. Coefficients of Equations Contain Cotangent......Page 178
6.7.1. Coefficients of Equations Contain Arcsine......Page 179
6.7.2. Coefficients of Equations Contain Arccosine......Page 180
6.7.4. Coefficients of Equations Contain Arccotangent......Page 181
6.8.1. Coefficients of Equations Contain Arbitrary Functions of x......Page 182
6.8.2. Coefficients of Equations Contain Arbitrary Functions of Different Variablcs ........Page 184
6.8.3. Coefficients of Equations Contain Arbitrary Functions of Two Variables......Page 185
7.1.1. Solution Methods......Page 188
7.1.2. Examples......Page 189
7.2.1. Coefficients of Equations Are Linear in x, y, and z......Page 190
7.2.2. Coefficients of Equations Are Quadratic in x, y, and z......Page 191
7.2.3. Coefficients of Equations Contain Other Powers in x, y, and z......Page 192
7.2.4. Coefficients of Equations Contain Arbitrary Powers of x, y, and z......Page 193
7.3.1. Coefficients of Equations Contain Exponential Functions......Page 195
7.3.2. Cocfficients of Equations Contain Exponential and Power-Law Functions......Page 196
7.4.1. Coefficients of Equations Contain Hyperbolic Sine......Page 197
7.4.2. Coefficients of Equations Contain Hyperbolic Cosine......Page 198
7.4.3. Coefficients of Equations Contain Hyperbolic Tangent......Page 199
7.4.4. Coefficients of Equations Contain Hyperbolic Cotangent......Page 200
7.5.1. Coefficicnts of Equations Contain Logarithmic Functions......Page 201
7.6.1. Coefficients of Equations Contain Sine......Page 202
7.6.2. Coefficients of Equations Contain Cosine......Page 203
7.6.3. Coefficients of Equations Contain Tangent......Page 204
7.6.5. Coefficients of Equations Contain Different Trigonometric Functions......Page 205
7.7.1. Coefficients of Equations Contain Arcsine......Page 206
7.7.3. Coefficients of Equations Contain Arctangent......Page 207
7.8.1. Coefficients of Equations Contain Arbitrary Functions of x......Page 208
7.8.2. Coefficients of Equations Contain Arbitrary Functions of Different Variables ........Page 210
7.8.3. Coefficients of Equations Contain Arbitrary Functions of Two Variables......Page 211
8.l.1. Solution Methods......Page 214
8.1.2. Examples......Page 215
8.2.1. Coefficients of Equations Are Linear in x, y, and z......Page 216
8.2.2. Coefficients of Equations Are Quadratic in x, y, and z......Page 217
8.2.3. Coefficients of Equations Contain Other Powers of x, y, and z......Page 218
8.2.4. Coefficients of Equations Contain Arbitrary Powers of x, y, and z......Page 219
8.3.1. Coefficients of Equations Contain Exponential Functions......Page 221
8.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions......Page 222
8.4.1. Coefficients of Equations Contain Hyperbolic Sine......Page 223
8.4.2. Coefficients of Equations Contain Hyperbolic Cosine......Page 224
8.4.3. Coefficients of Equations Contain Hyperbolic Tangent......Page 225
8.4.5. Coefficients of Equations Contain Different Hyperbolic Functions......Page 226
8.5.1. Coefficients of Equations Contain Logarithmic Functions......Page 227
8.6.1. Coefficients of Equations Contain Sine......Page 228
8.6.2. Coefficients of Equations Contain Cosine......Page 229
8.6.4. Coefficients of Equations Contain Cotangent......Page 230
8.6.5. Coefficients of Equations Contain Different Trigonometric Functions......Page 231
8.7.2. Coefficients of Equations Contain Arccosine......Page 232
8.7.4. Coefficients of Equations Contain Arccotangent......Page 233
8.8.1. Coefficients of Equations Contain Arbitrary Functions of x......Page 234
8.8.2. Coefficients of Equations Contain Arbitrary Functions of Different Variables ........Page 236
8.8.3. Coefficients of Equations Contain Arbitrary Functions of Two Variables......Page 237
9.1.1. Solution Methods......Page 240
9.2.1. Coefficients of Equations Are Linear in x, y, and z......Page 241
9.2.3. Coefficients of Equations Contain Other Powers of x, y, and z......Page 243
9.2.4. Coefficients of Equations Contain Arbitrary Powers of x, y, and z......Page 244
9.3.1. Coefficients of Equations Contain Exponential Functions......Page 246
9.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions......Page 247
9.4.1. Coefficients of Equations Contain Hyperbolic Sine......Page 248
9.4.2. Coefficients of Equations Contain Hyperbolic Cosine......Page 249
9.4.4. Coefficients of Equations Contain Hyperbolic Cotangent......Page 250
9.4.5. Coefficients of Equations Contain Different Hyperbolic Functions......Page 251
9.5.1. Coefficients of Equations Contain Logarithmic Functions......Page 252
9.6.1. Coefficients of Equations Contain Sine......Page 253
9.6.2. Coefficients of Equations Contain Cosine......Page 254
9.6.4. Coefficients of Equations Contain Cotangent......Page 255
9.6.5. Coefficients of Equations Contain Different Trigonometric Functions......Page 256
9.7.2. Coefficients of Equations Contain Arccosine......Page 257
9.7.4. Coefficients of Equations Contain Arccotangent......Page 258
9.8.1. Coefficients of Equations Contain Arbitrary Functions of x......Page 259
9.8.2. Coefficients of Equations Contain Arbitrary Functions of Different Variables ........Page 260
9.8.3. Coefficients of Equations Contain Arbitrary Functions of Two Variables......Page 261
10.1.1. Linear Homogeneous Equations......Page 264
10.1.2. Linear Nonhomogeneous Equations......Page 265
10.2.1. Equations Containing Power-Law Functions......Page 266
10.2.2. Other Equations Containing Arbitrary Parameters......Page 269
10.2.3. Equations Containing Arbitrary Functions......Page 271
Part III. Nonlinear Equations......Page 274
11.1.1. Solution Methods......Page 275
11.1.2. Examples......Page 276
11.2.1. Coefficients of Equations Contain Power-Law Functions......Page 277
11.2.2. Coefficients of Equations Contain Exponential Functions......Page 278
11.2.3. Coefficients of Equations Contain Hyperbolic Functions......Page 280
11.2.4. Coefficients of Equations Contain Logarithmic Functions......Page 281
11.3.1. Equations Contain Arbitrary Functions of One Variable......Page 282
11.3.2. Equations Contain Arbitrary Functions of Two Variables......Page 285
12.1.1. Solution Methods......Page 287
12.1.2. Cauchy Problem. Existence and Uniqueness Theorem......Page 288
12.1.3. Equation + few) = O. Qualitative Features and Discontinuous Solutions......Page 290
12.1.4. Generalized Solutions of Quasilinear Equations......Page 300
12.2.1. Coefficients of Equations Are Linear in w......Page 304
12.2.2. Coefficients of Equations Are Quadratic in w......Page 307
12.2.3. Coefficients of Equations Contain Other Powers of w......Page 309
12.3.1. Coefficients of Equations Contain Exponential Functions......Page 311
12.3.2. Coefficients of Equations Contain Hyperbolic Functions......Page 313
12.3.3. Coefficients of Equations Contain Logarithmic Functions......Page 316
12.3.4. Coefficients of Equations Contain Trigonometric Functions......Page 318
12.4.1. Equations Contain Arbitrary Functions of Independent Variables......Page 320
12.4.2. Equations Contain Arbitrary Functions of the Unknown Variable......Page 324
12.4.3. Equations Contain Arbitrary Functions of Two Variables......Page 328
13.2.1. Equations of the Fonn = f(x, y, w)......Page 331
13.2.2. Equations of the Form lex, y, w) + g(x, y, w) = h(x, y, w)......Page 333
13.2.3. Equations of the Fonn f(x, y, w) ; + g(x, y, W) + h(x, y, w) = s(x, y, w)......Page 335
13.2.4. Equations of the Form + lex, y, w){ ; )2 = g(x, y, w)......Page 338
13.2.5. Equations of the Form + lex, y, w)( )2 + g(x, y, w) = h(x, y, w)......Page 345
13.2.6. Equations of the Form f(x, y, w)( )2 + g(x, y, w){ )2 = h(x, y, w)......Page 349
13.2.7. Equations of the Form lex, y)( ) 2 + g(x, y) = h(x, y, w)......Page 354
13.2.8. Other Equations......Page 357
13.3.1. Equations of the Form = f(x, y, w)......Page 361
13.3.2. Equations of the Fonn f(x, y) + g(x, y) = h(x, y, w)......Page 363
13.3.3. Equations of the Form l(x,y) + g(x,y,w)( r = h(x,y,w)......Page 365
13.3.4. Equations of the Form + f(x, y, w)( ) 2 + g(x, y, w) = h(x, y, w)......Page 370
13.3.5. Equations of the Form f(x, y, w){ F- + g(x, y, w){ ) 2 = h(x, y, w)......Page 373
13.3.6. Equations of the Fonn ( )2 + f(x, y, w) = g(x, y, w)......Page 375
13.3.7. Other Equations......Page 378
14.1.1. Solution Methods......Page 381
14.1.2. Cauchy Problem. Existence and Uniqueness Theorem......Page 385
14.1.3. Generalized Viscosity Solutions and Their Applications......Page 388
14.2.1. Equations of the Form ( )2 = f(x, y, w)......Page 392
14.2.2. Equations of the Form f(x, y, w)( )3 + g(x, y, w) ; = h(:!;, y, w)......Page 393
14.2.3. Equations of the Form f(x, y, w)( )3 + g(x, y, w)( )1 = h(x, y, w)......Page 395
14.2.4. Equations of the Form f(x, y, w)( )3 + g(x, y, w) ; = h(x, y, w)......Page 396
14.2.5. Other Equations......Page 397
14.3.1. Equations Contain the Fourth Powers of Derivatives......Page 399
14.3.3. Equations Contain Arbitrary Powers of Derivatives......Page 401
14.3.4. More Complicated Equations......Page 404
14.4.1. Equations Contain One Arbitrary Power of Derivative......Page 406
14.4.2. Equations Contain Two or Three Arbitrary Powers of Derivatives......Page 409
14.5.1. Equations Contain Arbitrary Functions of One Variable......Page 411
14.5.2. Equations Contain Arbitrary Functions of Two Variables......Page 414
14.5.3. Equations Contain Arbitrary Functions of Three Variables......Page 418
14.5.4. Equations Contain Arbitrary Functions of Four Variables......Page 420
15.1.1. Quasilincar Equations......Page 421
15.1.2. Nonlinear Equations......Page 423
15.1.3. Generalized Viscosity Solutions......Page 430
15.2.1. Equations Vith Three Variables......Page 432
15.2.2. Equations Vith Arbitrary Number of Variables......Page 436
15.3.1. Equations Contain Squares of One or Two Derivatives......Page 438
15.3.2. Equations Contain Squares of Three Derivatives......Page 443
15.3.3. Equations Contain Products of Derivatives With Respect to Different Variables......Page 444
15.4.1. Equations Cubic in Derivatives......Page 446
15.4.2. Equations Contain Roots and Moduli of Derivatives......Page 447
15.4.3. Equations Contain Arbitrary Powers of Derivatives......Page 448
15.5,1. Equations Quadratic in Derivatives......Page 451
15.5.2. Equations With Power Nonlinearity in Derivatives......Page 457
15.5.3. Equations With Arbitrary Dependence on Derivatives......Page 459
15.5.4. Nonlinear Equations of General Form......Page 460
15.6.1. Equations Quadratic in Derivatives......Page 464
15.6.2. Equations Contain Power-Law Functions of Derivatives......Page 466
15.7.1. Equations Quadratic in Derivatives......Page 468
15.7.2. Equations With Power-Law Nonlinearity in Derivatives......Page 470
15.8.1. Equations Quadratic in Derivatives......Page 471
15.8.2. Equations With Power-Law Nonlinearity in Derivatives......Page 476
15.8.3. Equations Contain Arbitrary Functions of Two Variables......Page 477
15.8.4. Nonlinear Equations of General Fonn......Page 478
S.1.2. Reduce Notation Used in CONVODE......Page 483
S.1.3. How CONVODE Solves Equations......Page 484
S.2.1. Riccati Equation (Example 1)......Page 485
S.2.2. Riccati Equation (Example 2)......Page 489
S.2.3. A Nonlinear Equation Quadratic in the Derivative......Page 493
S.3.1. A First Order Linear Equation (Example 1)......Page 495
S.3.2. A First Order Linear Equation (Example 2)......Page 497
S.3.3. A Second Order Nonlinear Equation......Page 500
S.4.1. Arguments of the CONVODE procedure......Page 504
S.4.2. Global variables......Page 505
S.4.3. CONVODE via e-mail......Page 506
References......Page 507
Index......Page 511
