ODE-ProjectODE-Project Notation

Appendix C Notation

The following table defines the notation used in this book. Page numbers or references refer to the first appearance of each symbol.

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Symbol	Description	Location
$x^{'} (t) = f (t, x), x (0) = x_{0}$	first-order initial value problem	Subsection 1.1.1
$\frac{d P}{d t} = k (1 - \frac{P}{N}) P$	logistic population model	Subsection 1.1.2
$m x^{″} + b x^{'} + k x = 0$	simple damped harmonic oscillator	Subsection 1.1.3
$\frac{d y}{d x} = M (x) N (y)$	separable differential equation	Subsection 1.2.1
$\frac{d x}{d t} + p (t) x = q (t)$	first-order linear differential equation	Subsection 1.5.2
$\frac{d x}{d t} = f_{λ} (x)$	one-parameter family	Subsection 1.7.2
$x (t)$	vector-valued function	Subsection 2.2.2
$\frac{d x}{d t} = f (t, x), x (t_{0}) = x_{0}$	vector form of a system	Subsection 2.3.2
$\frac{d x}{d t} = f (x), \frac{d y}{d t} = g (x, y)$	partially coupled system	Subsection 2.4.1
$\frac{d x}{d t} = A x$	matrix notation for a system	Section 3.1
$A^{- 1}$	inverse of a matrix $A$	Subsection 3.1.1
$det (A)$	determinant of $A$	Subsection 3.1.1
$x^{T}$	matrix transpose	Subsection 3.1.2
$det (A - λ I) = λ^{2} - (a + d) λ + (a d - b c)$	characteristic polynomial	Subsection 3.1.3
$tr (A)$	trace of $A$	Exercises 3.1.6
$I$	identity matrix	Exercises 3.1.6
$e^{i β t} = \cos β t + i \sin β t$	Euler’s formula	Subsection 3.4.1
$x_{Re}$	real part of a complex number or vector	Subsection 3.4.1
$x_{Im}$	imaginary part of a complex number or vector	Subsection 3.4.1
$\overset{―}{λ}$	complex conjugate	Subsection 3.4.3
$e^{A}$	matrix exponential	Subsection 3.9.1
$e^{A}$	matrix exponential	Subsection 3.9.1
$a (t) x^{″} + b (t) x^{'} + c (t) x = g (t)$	second-order linear differential equation	Section 4.1
$p (λ) = det (A - λ I) = λ^{2} + \frac{b}{a} λ + \frac{c}{a}$	characteristic polynomial	Subsection 4.1.2
$W [f, g] (t)$	Wronskian	Exercises 4.2.7
$ω_{0}$	natural frequency	Subsection 4.4.1
$ω$	driving frequency	Subsection 4.4.1
$\overset{―}{ω}$	mean frequency	Subsection 4.4.2
$δ$	half difference	Subsection 4.4.2
$H (λ)$	transfer function	Subsection 4.4.3
$G (ω)$	gain	Subsection 4.4.3
$J$	Jacobian matrix	Subsection 5.1.1
$H$	Hamiltonian function	Subsection 5.2.2
$L (f) (s)$	Laplace transform	Subsection 6.1.1
$u_{a} (t) = u (t - a)$	Heaviside function	Example 6.1.5
$L^{- 1} (F (s)) (t)$	inverse Laplace transform	Subsection 6.1.3
$Γ (x)$	gamma function	Exercises 6.1.7
$δ$	Dirac delta function	Subsection 6.3.1
$f * g$	convolution product	Subsection 6.4.1

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