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Index

absolute convergence, 1
acceleration, 2
addition of matrices, 3
Airy’s equation, 4
algebraic multiplicity, 5
almost linear, 6
amplitude, 7
analytic functions, 8
angular frequency, 9
antiderivative, 10
antidifferentiate, 11
associated homogeneous equation, 12, 13
asymptotically stable, 14
asymptotically stable limit cycle, 15
atan, 16
atan2, 17
attractor, 18
augmented matrix, 19
autonomous, 20
autonomous equation, 21
autonomous system, 22
beating, 23

Bendixson-Dulac Theorem, 24
Bernoulli equation, 25
Bessel function of second kind, 26
Bessel function of the first kind, 27
Bessel’s equation, 28
boundary conditions for a PDE, 29
boundary value problem, 30
Burger’s equation, 31
catenary, 32

Cauchy-Euler equation, 33, 34
center, 35, 36
cgs units, 37, 38, 39
chaotic systems, 40
characteristic coordinates, 41
characteristic equation, 42
Chebyshev’s equation of order p , 43
Chebyshev’s equation of order 1, 44
closed curves, 45
closed trajectory, 46
coefficients, 47
cofactor, 48
cofactor expansion, 49
column vector, 50
commute, 51
complementary solution, 52
complete eigenvalue, 53
complex conjugate, 54
complex number, 55
complex roots, 56
conservative equation, 57
conservative vector field, 58
constant coefficient, 59, 60
constant coefficients, 61
constant of motion, 62
convergence of a power series, 63
convergent power series, 64
converges absolutely, 65
convolution, 66
corresponding eigenfunction, 67
cosine series, 68
critical point, 69, 70
critically damped, 71
d’Alembert solution to the wave equation, 72

damped, 73
damped motion, 74
damped nonlinear pendulum equation, 75
defect, 76
defective eigenvalue, 77
deficient matrix, 78
delta function, 79
dependent variable, 80
determinant, 81
diagonal matrix, 82
    matrix exponential of, 83
diagonalization, 84
differential equation, 85
Dirac delta function, 86
direction field, 87
Dirichlet boundary conditions, 88, 89
Dirichlet problem, 90
displacement vector, 91
distance, 92
divergent power series, 93
dot product, 94, 95
Duffing equation, 96
dynamic damping, 97
eigenfunction, 98, 99

eigenfunction decomposition, 100, 101
eigenvalue, 102, 103
eigenvalue of a boundary value problem, 104
eigenvector, 105
eigenvector decomposition, 106, 107
ellipses (vector field), 108
elliptic PDE, 109
endpoint problem, 110
envelope curves, 111
equilibrium, 112
equilibrium solution, 113, 114
Euler’s equation, 115
Euler’s formula, 116
Euler’s method, 117
Euler-Bernoulli equation, 118
even function, 119, 120
even periodic extension, 121
exact equation, 122
existence and uniqueness, 123, 124, 125
exponential growth model, 126
exponential of a matrix, 127
exponential order, 128
extend periodically, 129
first order differential equation, 130

first order linear equation, 131
first order linear system of ODEs, 132
first order method, 133
first shifting property, 134
forced motion, 135
    systems, 136
Fourier series, 137
fourth order method, 138
Fredholm alternative
    simple case, 139
    Sturm-Liouville problems, 140
free motion, 141
free variable, 142
Frobenius method, 143
Frobenius-type solution, 144
fundamental matrix, 145
fundamental matrix solution, 146, 147
general solution, 148

generalized eigenvectors, 149, 150
generalized function, 151
Genius software, 152
geometric multiplicity, 153
geometric series, 154, 155
Gibbs phenomenon, 156
half period, 157

Hamiltonian, 158
harmonic conjugate, 159
harmonic function, 160, 161
harvesting, 162
Heat equation, 163
heat equation, 164, 165
Heaviside function, 166
Hermite’s equation of order  , 167
Hermite’s equation of order 2, 168
homogeneous, 169
homogeneous equation, 170
homogeneous linear equation, 171
homogeneous side conditions, 172
homogeneous system, 173
Hooke’s law, 174, 175
hyperbolic PDE, 176
identity matrix, 177

imaginary part, 178
implicit solution, 179
Improved Euler’s method, 180
impulse response, 181, 182
inconsistent system, 183
indefinite integral, 184
independent variable, 185
indicial equation, 186
inhomogeneous, 187
initial condition, 188
initial conditions for a PDE, 189
inner product, 190
inner product of functions, 191, 192
integral equation, 193, 194
integrate, 195
integrating factor, 196
integrating factor method, 197
    for systems, 198
inverse Laplace transform, 199
invertible matrix, 200
IODE software, 201
isolated critical point, 202
Jacobian matrix, 203

la vie, 204

Laplace equation, 205, 206
Laplace transform, 207
Laplacian, 208
Laplacian in polar coordinates, 209
leading entry, 210
Leibniz notation, 211, 212
limit cycle, 213
linear combination, 214, 215
linear equation, 216, 217, 218
linear first order system, 219
linear operator, 220, 221
linear PDE, 222
linearity of the Laplace transform, 223
linearization, 224
linearly dependent, 225
linearly independent, 226, 227
    for vector valued functions, 228
logistic equation, 229
    with harvesting, 230
Lorenz attractor, 231
Lorenz system, 232
Lotka-Volterra, 233
mass matrix, 234

mathematical model, 235
mathematical solution, 236
matrix, 237
matrix exponential, 238
matrix inverse, 239
matrix valued function, 240
Maxwell’s equations, 241
Mechanical vibrations, 242
Method of Frobenius, 243
method of partial fractions, 244
Mixed boundary conditions, 245
mks units, 246, 247, 248
multiplication of complex numbers, 249
multiplicity, 250
multiplicity of an eigenvalue, 251
natural (angular) frequency, 252

natural frequency, 253, 254
natural mode of oscillation, 255
Neumann boundary conditions, 256, 257
Newton’s law of cooling, 258, 259, 260, 261
Newton’s second law, 262, 263, 264, 265
nilpotent, 266
nonhomogeneous, 267
nonlinear equation, 268
normal mode of oscillation, 269
odd function, 270, 271

odd periodic extension, 272
ODE, 273, 274
one-dimensional heat equation, 275
one-dimensional wave equation, 276
operator, 277
order, 278
ordinary differential equation, 279
Ordinary differential equations, 280
ordinary point, 281
orthogonal
    functions, 282, 283
    vectors, 284
    with respect to a weight, 285
orthogonality, 286
overdamped, 287
parabolic PDE, 288

parallelogram, 289
partial differential equation, 290, 291
Partial differential equations, 292
partial sum, 293
particular solution, 294, 295
PDE, 296, 297, 298
period, 299
periodic, 300
periodic orbit, 301
phase diagram, 302, 303
phase plane portrait, 304
phase portrait, 305, 306, 307
phase shift, 308
Picard’s theorem, 309
piecewise continuous, 310
piecewise smooth, 311
Poincarè section, 312
Poincarè-Bendixson Theorem, 313
Poisson kernel, 314
potential function, 315
power series, 316
practical resonance, 317, 318
practical resonance amplitude, 319
practical resonance frequency, 320
predator-prey, 321
product of matrices, 322
projection, 323
proper rational function, 324
pseudo-frequency, 325
pure resonance, 326, 327
quadratic formula, 328

radius of convergence, 329

ratio test for series, 330
real part, 331
real world problem, 332
recurrence relation, 333
reduced row echelon form, 334
reduction of order method, 335
regular singular point, 336
regular Sturm-Liouville problem, 337
reindexing the series, 338
relaxation oscillation, 339
repeated roots, 340
resonance, 341, 342, 343, 344
RLC circuit, 345
Robin boundary conditions, 346
row vector, 347
Runge-Kutta method, 348
saddle point, 349

sawtooth, 350
scalar, 351
scalar multiplication, 352
second order differential equation, 353
second order linear differential equation, 354
second order method, 355
second shifting property, 356
semistable critical point, 357
separable, 358
separation of variables, 359
shifting property, 360, 361
side conditions for a PDE, 362
Sierpinski triangle, 363
simple harmonic motion, 364
simply connected region, 365
sine series, 366
singular matrix, 367
singular point, 368
singular solution, 369
sink, 370
slope field, 371
solution, 372
solution curve, 373
source, 374
spiral sink, 375
spiral source, 376
square wave, 377, 378
stable center, 379
stable critical point, 380, 381
stable node, 382
steady periodic solution, 383, 384
steady state temperature, 385, 386
stiff problem, 387
stiffness matrix, 388
strange attractor, 389
Sturm-Liouville problem, 390
    regular, 391
superposition, 392, 393, 394, 395
symmetric matrix, 396, 397
system of differential equations, 398, 399
Taylor series, 400

tedious, 401, 402, 403, 404, 405
thermal conductivity, 406
three mass system, 407
three-point beam bending, 408
timbre, 409
total derivative, 410
trajectory, 411
transfer function, 412
transient solution, 413
Transport equation, 414
transpose, 415
trigonometric series, 416
undamped, 417

undamped motion, 418
    systems, 419
underdamped, 420
undetermined coefficients, 421
    for second order systems, 422, 423
    for systems, 424
unforced motion, 425
unit step function, 426
unstable critical point, 427, 428
unstable node, 429
upper triangular matrix, 430
Van der Pol oscillator, 431

variation of parameters, 432
    for systems, 433
vector, 434
vector field, 435, 436, 437
vector valued function, 438
velocity, 439
Volterra integral equation, 440
wave equation, 441, 442, 443

Wave equation in 2 dimensions, 444
weight function, 445