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Index Index

absolute convergence, Paragraph
acceleration, Paragraph
adding vectors, Paragraph
addition of matrices, Paragraph
Airy’s equation, Example
algebraic multiplicity, Paragraph
almost linear, Paragraph
amplitude, Paragraph
analytic functions, Paragraph
angular frequency, Paragraph
ansatz, Paragraph
antiderivative, Paragraph
antidifferentiate, Paragraph
associated homogeneous equation, Paragraph Theorem
associative law, Paragraph
asymptotically stable, Paragraph
asymptotically stable limit cycle, Paragraph
atan, Paragraph
atan2, Paragraph
attractor, Paragraph
augmented matrix, Paragraph Paragraph
autonomous, Paragraph
autonomous equation, Paragraph
autonomous system, Paragraph
backsubstitution, Paragraph
beating, Example
Bendixson–Dulac Theorem, Theorem
Bernoulli equation, Paragraph
Bessel function of second kind, Paragraph
Bessel function of the first kind, Paragraph
Bessel’s equation, Paragraph
boundary conditions for a PDE, Paragraph
boundary value problem, Paragraph
Burger’s equation, Paragraph
cartesian plane, Paragraph
catenary, Paragraph
Cauchy–Euler equation, Paragraph Exercise
chaotic systems, Paragraph
characteristic coordinates, Example Paragraph
characteristic curve, Example
characteristic equation, Paragraph
characteristics, Paragraph
Chebyshev’s equation of order 1, Exercise
Chebyshev’s equation of order \(p\), Exercise
clamped end of beam, Paragraph
closed curves, Example
closed trajectory, Paragraph
coefficients, Paragraph
cofactor, Paragraph Paragraph
cofactor expansion, Paragraph Paragraph
column space, Paragraph
column vector, Paragraph Paragraph
commute, Item
complementary solution, Paragraph
complete eigenvalue, Paragraph
complex conjugate, Paragraph
complex number, Paragraph
complex roots, Paragraph
conservative equation, Paragraph
conservative vector field, Paragraph
consistent, Paragraph
constant coefficient, Paragraph Paragraph Paragraph
constant of motion, Paragraph
convection, Example
convection equation, Example
convergence of a power series, Paragraph
convergent power series, Paragraph
converges absolutely, Paragraph
convolution, Paragraph
corresponding eigenfunction, Paragraph
cosine series, Paragraph
critical point, Paragraph Paragraph
critically damped, Paragraph
damped, Paragraph
damped motion, Item
damped nonlinear pendulum equation, Exercise
defect, Paragraph
defective eigenvalue, Paragraph
deficient matrix, Paragraph
delta function, Paragraph
dependent variable, Paragraph
determinant, Paragraph Paragraph
diagonal matrix, Paragraph Paragraph
matrix exponential of, Paragraph
diagonalization, Paragraph
differential equation, Paragraph
Dirac delta function, Paragraph
direction, Paragraph Paragraph
direction field, Paragraph
Dirichlet boundary conditions, Paragraph Paragraph
Dirichlet problem, Paragraph
displacement vector, Paragraph
distance, Paragraph
distributive law, Paragraph
divergent power series, Paragraph
Duffing equation, Paragraph
dynamic damping, Exercise
d’Alembert solution to the wave equation, Paragraph
echelon form, Paragraph
eigenfunction, Paragraph Paragraph
eigenfunction decomposition, Paragraph Paragraph
eigenvalue, Paragraph Paragraph
eigenvalue of a boundary value problem, Paragraph
eigenvector, Paragraph
eigenvector decomposition, Subsubsection Paragraph
element of a matrix, Paragraph Paragraph
elementary operations, Paragraph
elementary row operations, Paragraph
elimination, Paragraph
ellipses (vector field), Paragraph
elliptic PDE, Paragraph
endpoint problem, Paragraph
entry of a matrix, Paragraph Paragraph
envelope curves, Paragraph
equilibrium, Paragraph
equilibrium solution, Paragraph Paragraph
euclidean space, Paragraph
Euler–Bernoulli equation, Paragraph
Euler’s equation, Paragraph
Euler’s formula, Paragraph
Euler’s method, Paragraph
even function, Example Paragraph
even periodic extension, Paragraph
exact equation, Paragraph
existence and uniqueness, Theorem Theorem Theorem
existence and uniqueness for systems, Theorem
exponential growth, Paragraph
exponential growth model, Paragraph
exponential of a matrix, Paragraph
exponential order, Paragraph
extend periodically, Paragraph
first order differential equation, Paragraph
first order linear equation, Paragraph
first order linear system of ODEs, Paragraph
first order method, Paragraph
first order system, Paragraph
first shifting property, Paragraph
fixed end of beam, Paragraph
forced motion, Item
systems, Paragraph
four fundamental equations, Paragraph
Fourier series, Paragraph
fourth order method, Paragraph
Fredholm alternative, Paragraph
simple case, Theorem
Sturm–Liouville problems, Theorem
free end of beam, Paragraph
free motion, Item
free variable, Paragraph Paragraph
frequency, Paragraph
Frobenius method, Paragraph
Frobenius-type solution, Theorem
fundamental frequency, Paragraph Paragraph
fundamental matrix, Paragraph
fundamental matrix solution, Paragraph Paragraph
Gaussian elimination, Paragraph
Gauss–Jordan elimination, Paragraph
general solution, Paragraph
general solution to a system, Paragraph
generalized eigenvectors, Example Paragraph
generalized function, Paragraph
Genius software, Paragraph
geometric multiplicity, Paragraph
geometric series, Paragraph Paragraph
Gibbs phenomenon, Paragraph
Gram–Schmidt process, Paragraph
half period, Paragraph
Hamiltonian, Paragraph
harmonic conjugate, Item
harmonic function, Exercise Paragraph
harvesting, Paragraph
heat equation, Paragraph Paragraph Paragraph
Heaviside function, Example
Hermite’s equation of order 2, Exercise
Hermite’s equation of order \(n\), Example
hinged end of beam, Paragraph
homogeneous, Paragraph
homogeneous equation, Paragraph
homogeneous linear equation, Paragraph
homogeneous matrix equation, Paragraph
homogeneous side conditions, Paragraph
homogeneous system, Paragraph
Hooke’s law, Paragraph Paragraph
hyperbolic cosine, Paragraph
hyperbolic PDE, Paragraph
hyperbolic sine, Paragraph
identity matrix, Paragraph Paragraph
ill-posed, Paragraph
imaginary part, Paragraph
implicit solution, Paragraph
Improved Euler’s method, Paragraph
impulse response, Paragraph Paragraph
inconsistent, Paragraph
inconsistent system, Paragraph
indefinite integral, Paragraph
independent variable, Paragraph
indicial equation, Example
inhomogeneous, Paragraph
initial condition, Paragraph
initial condition for a PDE, Paragraph
initial condition for a system, Paragraph
initial conditions for a PDE, Paragraph
inner product, Paragraph Paragraph Paragraph
inner product of functions, Paragraph Paragraph
integral equation, Example Paragraph
integrate, Paragraph
integrating factor, Paragraph
integrating factor method, Paragraph
for systems, Paragraph
inverse Laplace transform, Paragraph
invertible matrix, Paragraph Paragraph
IODE software, Paragraph
isolated critical point, Paragraph
Jacobian matrix, Paragraph
kernel, Paragraph
la vie, Paragraph
Laplace equation, Paragraph Paragraph
Laplace transform, Paragraph
Laplacian, Paragraph
Laplacian in polar coordinates, Paragraph
leading entry, Paragraph Paragraph
Leibniz notation, Paragraph Paragraph
limit cycle, Paragraph
linear combination, Paragraph Paragraph Paragraph Paragraph
linear equation, Paragraph Paragraph Paragraph
linear first order system, Paragraph
linear mapping, Paragraph
linear operator, Paragraph Paragraph
linear PDE, Paragraph
linear systems, Paragraph
linearity of the Laplace transform, Theorem
linearization, Paragraph
linearly dependent, Paragraph Paragraph
linearly independent, Paragraph Paragraph Paragraph
for vector-valued functions, Paragraph
logistic equation, Paragraph
with harvesting, Paragraph
Lorenz attractor, Paragraph
Lorenz system, Paragraph
Lotka–Volterra, Paragraph
lower triangular, Paragraph
magnitude, Paragraph
mass matrix, Paragraph
mathematical model, Paragraph
mathematical solution, Paragraph
matrix exponential, Paragraph
matrix inverse, Paragraph Paragraph
matrix product, Paragraph
matrix-valued function, Paragraph
Maxwell’s equations, Paragraph
mechanical vibrations, Paragraph
method of characteristics, Paragraph
Method of Frobenius, Paragraph
method of partial fractions, Example
Mixed boundary conditions, Paragraph
multiplication of complex numbers, Paragraph
multiplicity, Example
multiplicity of an eigenvalue, Paragraph
\(n\)-dimensional space, Paragraph
natural (angular) frequency, Paragraph
natural frequency, Paragraph Example
natural mode of oscillation, Example
Neumann boundary conditions, Paragraph Paragraph
Newton’s law of cooling, Paragraph Example Exercise Paragraph
Newton’s second law, Paragraph Paragraph Example Paragraph
nilpotent, Paragraph
nonhomogeneous, Paragraph
nonlinear equation, Paragraph
normal mode of oscillation, Example
nullity, Paragraph
nullspace, Paragraph
odd function, Example Paragraph
odd periodic extension, Paragraph
one-dimensional heat equation, Paragraph
one-dimensional wave equation, Paragraph
operator, Paragraph Paragraph
order, Paragraph
ordinary differential equation, Paragraph
Ordinary differential equations, Item
ordinary point, Paragraph
orthogonal, Paragraph
functions, Theorem Paragraph
vectors, Paragraph
with respect to a weight, Paragraph
orthogonal basis, Paragraph
orthogonal projection, Paragraph
orthogonality, Paragraph
orthonormal basis, Paragraph
overdamped, Paragraph
overtones, Paragraph Paragraph
parabolic PDE, Paragraph
parallelogram, Paragraph Paragraph Paragraph
partial differential equation, Paragraph Paragraph
Partial differential equations, Item
partial sum, Paragraph
particular solution, Paragraph Paragraph
period, Paragraph
periodic, Paragraph
periodic extension, Paragraph
periodic orbit, Paragraph
phase diagram, Paragraph Paragraph
phase plane portrait, Paragraph
phase portrait, Paragraph Paragraph Paragraph
phase shift, Paragraph
Picard’s theorem, Theorem Theorem
piecewise continuous, Paragraph
piecewise smooth, Paragraph
pivot, Paragraph
Poincaré section, Paragraph
Poincaré–Bendixson Theorem, Theorem
Poisson kernel, Paragraph
potential function, Paragraph
power series, Paragraph
practical resonance, Paragraph Paragraph
practical resonance amplitude, Paragraph
practical resonance frequency, Paragraph
predator-prey, Paragraph
product of matrices, Paragraph Paragraph
projection, Paragraph
orthogonal, Paragraph
proper rational function, Paragraph
pseudo-frequency, Paragraph
pulse, Paragraph
pure resonance, Paragraph Paragraph
quadratic formula, Paragraph
radius of convergence, Theorem
rank, Paragraph
ratio test for series, Paragraph
real part, Paragraph
real-world problem, Paragraph
rectangular pulse, Paragraph
recurrence relation, Example
reduced row echelon form, Paragraph Paragraph
reduction of order method, Paragraph
regular singular point, Paragraph
regular Sturm–Liouville problem, Paragraph
reindexing the series, Example
relaxation oscillation, Example
repeated roots, Example
RLC circuit, Paragraph
Robin boundary conditions, Paragraph
root test for series, Paragraph
row echelon form, Paragraph
row reduction, Paragraph
row space, Paragraph
row vector, Paragraph Paragraph
Runge–Kutta method, Paragraph
saddle point, Paragraph
sawtooth, Example Example
scalar multiplication, Paragraph
scale a vector, Paragraph
second order differential equation, Paragraph
second order linear differential equation, Paragraph
second order method, Paragraph
second order system, Paragraph
second shifting property, Paragraph
semistable critical point, Paragraph
separable, Paragraph
separation of variables, Paragraph
shifting property, Paragraph Paragraph
side conditions for a PDE, Paragraph
Sierpinski triangle, Exercise
simple harmonic motion, Paragraph
simply connected region, Paragraph
sine series, Paragraph
singular matrix, Paragraph Paragraph
singular point, Paragraph
singular solution, Paragraph
sink, Paragraph
slope field, Paragraph
solution, Paragraph
solution curve, Paragraph
solution to a system, Paragraph
source, Paragraph
span, Paragraph
spectrum, Paragraph Paragraph
spiral sink, Paragraph
spiral source, Paragraph
square matrix, Paragraph
square wave, Paragraph Example
stable center, Paragraph
stable critical point, Paragraph Paragraph
stable node, Paragraph
standard basis vectors, Paragraph
standard inner product, Paragraph
steady periodic solution, Paragraph Paragraph
steady state temperature, Exercise Paragraph
stiff problem, Item
stiffness matrix, Paragraph
strange attractor, Paragraph
Sturm–Liouville problem, Paragraph
regular, Paragraph
subspace, Paragraph
subtracting vectors, Paragraph
symmetric matrix, Paragraph Paragraph Paragraph
system of differential equations, Paragraph Paragraph
Taylor series, Paragraph
thermal conductivity, Paragraph
three mass system, Paragraph
three-point beam bending, Subsection
total derivative, Paragraph
trajectory, Paragraph
transfer function, Paragraph
transformation, Paragraph
transient solution, Paragraph
transport equation, Paragraph Example Example
transpose, Paragraph Paragraph
trigonometric series, Paragraph
undamped, Paragraph
undamped motion, Item
systems, Paragraph
underdamped, Paragraph
undetermined coefficients, Subsection
for second order systems, Paragraph Paragraph
for systems, Paragraph
unforced motion, Item
unit step function, Example
unit vector, Paragraph
unstable critical point, Paragraph Paragraph
unstable node, Paragraph
upper triangular, Paragraph
upper triangular matrix, Paragraph
Van der Pol oscillator, Example
variation of parameters, Paragraph
for systems, Paragraph
vector-valued function, Paragraph Paragraph
velocity, Paragraph
Volterra integral equation, Paragraph
wave equation, Paragraph Paragraph Paragraph
wave equation in 2 dimensions, Paragraph
wavefronts, Paragraph
weight function, Paragraph
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