Page 894 - Advanced_Engineering_Mathematics o'neil

874 Index

Eigenvalues λ (Continued) Euler’s differential equation, 72–74
real distinct and opposite signs, Euler’s formula, 457–458
332–333 Euler’s method, 139–141, 143–144
repeated roots, 273 approximation scheme, 139–141
Sturm-Liouville problems and, modified with Taylor method,
506–511 143–144
symmetric matrices, 273–275 Even functions, 436–438
zero and nonzero values of, 267–268 Even permutation, 247
Eigenvectors (E), 267–276, 279–283, Exact equations, 21–25
308–312 Exactness test, 23–25
characteristic polynomials, 269–271 Existence and uniqueness theorem,
defined, 267 40–41
diagonalization of matrices using, Exponential functions, 684–688
279–283 Exponential matrix solutions, 316–318
eigenvalues λ and, 267–276 defined, 316
homogeneous linear system solutions Laplace transform for, 317–318
and, 308–312 MAPLE command for, 316
linear dependence and independence variation of parameters, method of
of, 272–273, 308–312 for, 317–318
nonzero multiples of, 267–268
orthogonality of, 247–275 F
symmetric matrices, 273–275 Faraday’s law, 411
Electrical circuits, 33, 70–71, 105–106, Filtering,103–104, 461–463, 481–482
321–327 Cesàro function Z(t), 461–463
first-order differential equation Cesàro sum σ(t), 461–463
applications for, 33 Dirac delta function δ(t),103–104,
Kirchhoff’s current and voltage laws, 481–482
33 Fourier series used for, 461–463
second-order differential equation Fourier transforms and, 481–482
applications for, 70–71 function Z, 461–462
solutions for, 321–327 Gauss filter, 463
spring equation for, 70–71 graphs for, 462–463
transient modeling using Dirac delta Hamming filter, 463
function d(t), 105–106 property,103–104, 481
Element of a matrix, 187 signals, 461–463
Elementary matrix, 199 First-order differential equations, 3–42
Elliptical cone, surface integrals of, 390 Bernoulli equation, 27–28
Equal matrices, 187 electrical circuits, applications for, 33
Equilibrium position, 61 exact equations, 21–25
Equipotential lines, 783–784 existence and uniqueness theorem,
Error function (erf), 631 40–41
Essential singularities, 730 homogeneous equation, 26–27
Euler constant γ , 538 initial value problems, 6–8, 40–41

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