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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 frequency, 145
fundamental matrix, 146
fundamental matrix solution, 147, 148
general solution, 149

generalized eigenvectors, 150, 151
generalized function, 152
Genius software, 153
geometric multiplicity, 154
geometric series, 155, 156
Gibbs phenomenon, 157
half period, 158

Hamiltonian, 159
harmonic conjugate, 160
harmonic function, 161, 162
harvesting, 163
Heat equation, 164
heat equation, 165, 166
Heaviside function, 167
Hermite’s equation of order  , 168
Hermite’s equation of order 2, 169
homogeneous, 170
homogeneous equation, 171
homogeneous linear equation, 172
homogeneous side conditions, 173
homogeneous system, 174
Hooke’s law, 175, 176
hyperbolic PDE, 177
identity matrix, 178

imaginary part, 179
implicit solution, 180
Improved Euler’s method, 181
impulse response, 182, 183
inconsistent system, 184
indefinite integral, 185
independent variable, 186
indicial equation, 187
inhomogeneous, 188
initial condition, 189
initial conditions for a PDE, 190
inner product, 191
inner product of functions, 192, 193
integral equation, 194, 195
integrate, 196
integrating factor, 197
integrating factor method, 198
    for systems, 199
inverse Laplace transform, 200
invertible matrix, 201
IODE software, 202
isolated critical point, 203
Jacobian matrix, 204

la vie, 205

Laplace equation, 206, 207
Laplace transform, 208
Laplacian, 209
Laplacian in polar coordinates, 210
leading entry, 211
Leibniz notation, 212, 213
limit cycle, 214
linear combination, 215, 216
linear equation, 217, 218, 219
linear first order system, 220
linear operator, 221, 222
linear PDE, 223
linearity of the Laplace transform, 224
linearization, 225
linearly dependent, 226
linearly independent, 227, 228
    for vector valued functions, 229
logistic equation, 230
    with harvesting, 231
Lorenz attractor, 232
Lorenz system, 233
Lotka-Volterra, 234
mass matrix, 235

mathematical model, 236
mathematical solution, 237
matrix, 238
matrix exponential, 239
matrix inverse, 240
matrix valued function, 241
Maxwell’s equations, 242
Mechanical vibrations, 243
Method of Frobenius, 244
method of partial fractions, 245
Mixed boundary conditions, 246
mks units, 247, 248, 249
multiplication of complex numbers, 250
multiplicity, 251
multiplicity of an eigenvalue, 252
natural (angular) frequency, 253

natural frequency, 254, 255
natural mode of oscillation, 256
Neumann boundary conditions, 257, 258
Newton’s law of cooling, 259, 260, 261, 262
Newton’s second law, 263, 264, 265, 266
nilpotent, 267
nonhomogeneous, 268
nonlinear equation, 269
normal mode of oscillation, 270
odd function, 271, 272

odd periodic extension, 273
ODE, 274, 275
one-dimensional heat equation, 276
one-dimensional wave equation, 277
operator, 278
order, 279
ordinary differential equation, 280
Ordinary differential equations, 281
ordinary point, 282
orthogonal
    functions, 283, 284
    vectors, 285
    with respect to a weight, 286
orthogonality, 287
overdamped, 288
overtones, 289
parabolic PDE, 290

parallelogram, 291
partial differential equation, 292, 293
Partial differential equations, 294
partial sum, 295
particular solution, 296, 297
PDE, 298, 299, 300
period, 301
periodic, 302
periodic extension, 303
periodic orbit, 304
phase diagram, 305, 306
phase plane portrait, 307
phase portrait, 308, 309, 310
phase shift, 311
Picard’s theorem, 312
piecewise continuous, 313
piecewise smooth, 314
Poincarè section, 315
Poincarè-Bendixson Theorem, 316
Poisson kernel, 317
potential function, 318
power series, 319
practical resonance, 320, 321
practical resonance amplitude, 322
practical resonance frequency, 323
predator-prey, 324
product of matrices, 325
projection, 326
proper rational function, 327
pseudo-frequency, 328
pure resonance, 329, 330
quadratic formula, 331

radius of convergence, 332

ratio test for series, 333
real part, 334
real world problem, 335
recurrence relation, 336
reduced row echelon form, 337
reduction of order method, 338
regular singular point, 339
regular Sturm-Liouville problem, 340
reindexing the series, 341
relaxation oscillation, 342
repeated roots, 343
resonance, 344, 345, 346, 347
RLC circuit, 348
Robin boundary conditions, 349
row vector, 350
Runge-Kutta method, 351
saddle point, 352

sawtooth, 353
scalar, 354
scalar multiplication, 355
second order differential equation, 356
second order linear differential equation, 357
second order method, 358
second shifting property, 359
semistable critical point, 360
separable, 361
separation of variables, 362
shifting property, 363, 364
side conditions for a PDE, 365
Sierpinski triangle, 366
simple harmonic motion, 367
simply connected region, 368
sine series, 369
singular matrix, 370
singular point, 371
singular solution, 372
sink, 373
slope field, 374
solution, 375
solution curve, 376
source, 377
spectrum, 378
spiral sink, 379
spiral source, 380
square wave, 381, 382
stable center, 383
stable critical point, 384, 385
stable node, 386
steady periodic solution, 387, 388
steady state temperature, 389, 390
stiff problem, 391
stiffness matrix, 392
strange attractor, 393
Sturm-Liouville problem, 394
    regular, 395
superposition, 396, 397, 398, 399
symmetric matrix, 400, 401
system of differential equations, 402, 403
Taylor series, 404

tedious, 405, 406, 407, 408, 409
thermal conductivity, 410
three mass system, 411
three-point beam bending, 412
timbre, 413
total derivative, 414
trajectory, 415
transfer function, 416
transient solution, 417
Transport equation, 418
transpose, 419
trigonometric series, 420
undamped, 421

undamped motion, 422
    systems, 423
underdamped, 424
undetermined coefficients, 425
    for second order systems, 426, 427
    for systems, 428
unforced motion, 429
unit step function, 430
unstable critical point, 431, 432
unstable node, 433
upper triangular matrix, 434
Van der Pol oscillator, 435

variation of parameters, 436
    for systems, 437
vector, 438
vector field, 439, 440, 441
vector valued function, 442
velocity, 443
Volterra integral equation, 444
wave equation, 445, 446, 447

Wave equation in 2 dimensions, 448
weight function, 449