# The Ordinary Differential Equations Project

## AppendixCNotation

The following table defines the notation used in this book. Page numbers or references refer to the first appearance of each symbol.
Symbol Description Location
$$x'(t) = f(t, x), x(0) = x_0$$ first-order initial value problem Subsection 1.1.1
$$\dfrac{dP}{dt} = k \left( 1 - \dfrac{P}{N} \right) P$$ logistic population model Subsection 1.1.2
$$mx'' + bx' + kx = 0$$ simple damped harmonic oscillator Subsection 1.1.3
$$\dfrac{dy}{dx} =M(x) N(y)$$ separable differential equation Subsection 1.2.1
$$\dfrac{dx}{dt} + p(t) x = q(t)$$ first-order linear differential equation Subsection 1.5.2
$$\dfrac{dx}{dt} = f_\lambda(x)$$ one-parameter family Subsection 1.7.2
$${\mathbf x}(t)$$ vector-valued function Subsection 2.2.2
$$\dfrac{d {\mathbf x}}{dt} = {\mathbf f}(t, {\mathbf x}), {\mathbf x}(t_0) = {\mathbf x}_0$$ vector form of a system Subsection 2.3.2
$$\dfrac{dx}{dt} = f(x),\dfrac{dy}{dt} = g(x, y)$$ partially coupled system Subsection 2.4.1
$$\dfrac{d \mathbf x}{dt} = A {\mathbf x}$$ 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
$$\mathbf x^T$$ matrix transpose Subsection 3.1.2
$$\det(A - \lambda I) = \lambda^2 - (a + d) \lambda + (ad - bc)$$ characteristic polynomial Subsection 3.1.3
$$\trace(A)$$ trace of $$A$$ Exercises 3.1.6
$$I$$ identity matrix Exercises 3.1.6
$$e^{i \beta t} = \cos \beta t + i \sin \beta t$$ Euler’s formula Subsection 3.4.1
$${ \mathbf x}_{\text{Re}}$$ real part of a complex number or vector Subsection 3.4.1
$${ \mathbf x}_{\text{Im}}$$ imaginary part of a complex number or vector Subsection 3.4.1
$$\overline{\lambda}$$ 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
$$p(\lambda) = \det(A - \lambda I) = \lambda^2 + \frac{b}{a} \lambda + \frac{c}{a}$$ characteristic polynomial Subsection 4.1.2
$$W[f, g](t)$$ Wronskian Exercises 4.2.7
$$\omega_0$$ natural frequency Subsection 4.4.1
$$\omega$$ driving frequency Subsection 4.4.1
$$\overline{\omega}$$ mean frequency Subsection 4.4.2
$$\delta$$ half difference Subsection 4.4.2
$$H(\lambda)$$ transfer function Subsection 4.4.3
$$G(\omega)$$ gain Subsection 4.4.3
$$J$$ Jacobian matrix Subsection 5.1.1
$$H$$ Hamiltonian function Subsection 5.2.2
$${\mathcal L}(f)(s)$$ Laplace transform Subsection 6.1.1
$$u_a(t) = u(t - a)$$ Heaviside function Example 6.1.5
$${\mathcal L}^{-1}(F(s))(t)$$ inverse Laplace transform Subsection 6.1.3
$$\Gamma(x)$$ gamma function Exercises 6.1.7
$$\delta$$ Dirac delta function Subsection 6.3.1
$$f*g$$ convolution product Subsection 6.4.1