Index

A

Abel’s formula, 5.1.4–5.1.6, 9.1.3
Accelerated payment, 4.1.23
Acceleration due to gravity, 4.3.2
Airy’s equation, 7.2.3
Amplitude
- of oscillation, 6.1.10
- time-varying, 6.2.3
Amplitude–phase form, 6.1.2
Aphelion distance, 6.4.11
Apogee, 6.4.11
Applications
- of first order equations, 4.1.1–4.5.34
  - autonomous second order equations, 4.4.1–4.5.1
  - cooling problems, 4.2.1–4.2.4
  - curves, 4.5.1–4.5.34
  - elementary mechanics, 4.3.1–4.4.20
  - growth and decay, 4.1.1–4.1.27
  - mixing problems, 4.2.3–4.2.23
- of linear second order equations, 6.1.1–6.4.6
  - motion under a central force, 6.3.17–6.4.6
  - motion under inverse square law force, 6.4.10–6.4.1
  - \(RLC\) circuit, 6.2.26–6.3.17
  - spring–mass systems, 6.1.1–6.2.26
Autonomous second order equations, 4.4.1–4.5.7
- conversion to first order equations, 4.4.1
- damped 4.4.26–4.4.24
  - pendulum 4.4.5
  - spring–mass system, 4.4.26
- Newton’s second law of motion and, 4.4.6
- undamped 4.4.7–4.4.24
  - pendulum 4.4.26–4.4.17
  - spring–mass system, 4.4.8–4.4.26
- stability and instability conditions for, 4.4.18–4.4.24

B

D

D’Alembert’s solution, 12.2.1
Damped autonomous second order equations, 4.4.24–4.5.1
- for pendulum, 4.4.5
- for spring-mass system, 4.4.26
Damped motion, 6.1.1
Damping
- \(RLC\) circuit in forced oscllation with, 6.3.17
- spring-mass systems with, 4.4.26, 6.1.1, 6.2.1–6.2.25
  - critically damped motion, 6.2.1–6.2.11
  - forced vibrations, 6.2.11–6.2.1
  - free vibrations, 6.2.1–6.2.11
  - overdamped motion,6.2.3
  - underdamped motion, 6.2.3
- spring-mass systems without, 6.1.1–6.1.1
  - forced oscillation, 6.1.13–6.1.1
Damping constant, 4.4.26
Damping forces, 4.4.6, 6.1.1
Dashpot, 6.1.1
Dating, carbon, 4.1.12–4.1.6
Death rate 1.1.3
Decay, See Exponential growth and decay
Decay constant, 4.1.3
Derivatives, Laplace transform of, 8.3.1–8.3.2
Differential equations
- defined, 1.2.1
- order of, 1.2.2
- ordinary, 1.2.1
- partial, 1.2.1
- solutions of, 1.2.1–1.2.5
Differentiation of power series, 7.1.4
Dirac, Paul A. M., 8.6.14
Dirac delta function, 8.6.14
Direction fields for first order equations, 1.3.1–1.3.24
Dirichlet, Peter G. L., 12.3.2
Dirichlet condition, 12.3.2
Dirichlet problem, 12.3.2
Discontinuity
- jump, 8.1.12
- removable, 8.2.12
Distributions, theory of, 8.7.1
Divergence of improper integral, 8.1.1
Divergent power series, 7.1.1

E

Eccentricity of orbit, 6.4.11
Eigenfunction associated with \(\lambda\), 11.1.1, 13.2.1
Eigenvalue, 11.1.1, 13.2.1
Eigenvalue problems, See also Boundary value problems
- Sturm-Liouville problems, 13.1.32–13.2.32
  - defined, 13.2.10
  - orthogonality in, 13.2.5
  - solving, 13.2.2
Elliptic orbit, 6.4.11
Epidemics 1.1.8–2.2.29
Equidimensional equation, 9.2.44
Equilibrium, 4.4.6
- spring-mass system, 6.1.1
Equilibrium position, 6.1.1
Equipotentials, 4.5.16
Error(s)
- in applying numerical methods, 3.1.1
- in Euler’s method, 3.1.2–3.1.18
- at the \(i\)-th step, 3.1.1
- truncation, 3.1.1
  - global, 3.1.18, 3.2.5, 3.3.1
  - local, 3.1.8
  - local, numerical methods with \(O(h^{3})\), 3.2.8–3.2.1
Escape velocity, 4.3.20
Euler’s equation, 7.4.2–7.4.20, 5.3.34
Euler’s identity, 3.1.2–3.1.22
Euler’s method, 3.1.2–3.1.23
- error in, 3.1.2–3.1.8
  - truncation, 3.1.8–3.1.18
- improved, 3.2.1–3.2.7
- semilinear, 3.1.19–3.1.1
- step size and accuracy of, 3.1.2
Even functions, 11.2.5
Exact first order equations, 2.5.1–2.5.44
- implicit solutions of, 2.5.1–2.5.1
- procedurs for solving, 2.5.19
Exactness condition, 2.5.2
Existence of solutions of nonlinear first order equations, 2.3.1–2.3.21
Existence theorem, 2.1.2, 2.3.1
Exponential growth and decay, 4.1.1–4.1.27
- carbon dating, 4.1.5
- interest compounded continuously, 4.1.2
- mixed growth and decay, 4.1.10
- radioactive decay, 4.1.3
- savings program, 4.1.6
Exponential order, function of, 8.1.5

F

G

Gamma function, 8.1.16
Generalized Riccati equation, 2.4.55, 5.6.40
General solution
- of higher order constant coefficient homogeneous equations, 9.2.1–9.2.1
- of homogeneous linear second order equations, 5.1.2
- of homogeneous linear systems of differential equations, 10.3.1, 10.3.3
- of linear higher order equations, 9.1.3, 9.1.5
- of nonhomogeneous linear first order equations, 2.1.2, 2.1.2
- of nonhomogeneous linear second order equations, 5.3.5, 5.6.1–5.6.40
Geometric problems, 5.4.7–4.5.8
Gibbs phenomenon, 11.2.1, 11.2.7
Global truncation error in Euler’s method, 3.1.18
Glucose absorption by the body, 1.1.7
Gravitation, Newton’s law of, 4.3.1, 4.4.17, 6.3.17, 10.1.3, 10.1.4
Gravity, acceleration due to,4.3.1
Green’s function, 9.4.34, 13.1.17, 13.1.6, 13.1.21–13.1.25
Grid, rectangular, 1.3.1
Growth and decay
- carbon dating, 4.1.5
- exponential, 4.1.1–4.1.27
- interest compounded continuously, 4.1.8–4.1.3
- mixed growth and decay, 4.1.4
- radioactive decay, 4.1.3
- savings program, 4.1.6

H

I

Imaginary part, 5.2.17
Implicit function theorem, 2.2.1
Implicit solution(s) 2.5.3–2.5.8
- of exact first order equations, 2.5.3–2.5.8
- of initial value problems, 2.2.1
- of separable first order equations, 2.2.1–2.2.5
Impressed voltage, 2.2.26
Improper integral, 8.1.1
Improved Euler method, 3.1.23–3.2.4 3.3.2–3.3.4
- semilinear, 3.2.4–3.2.7
Impulse function, 8.6.14
Impulse response, 8.6.1, 8.7.2
Impulses, constant coefficient equations with, 8.6.14–8.7.30
Independence, linear
- of \(n\) function, 9.1.2
- of two functions, 5.1.3
- of vector functions, 10.3.2
Indicial equation, 7.4.7, 7.5.3
- with distinct real roots differing by an integer, 7.7.1–7.7.48
- with distinct real roots not differing by an integer, 7.5.3–7.5.68
- with repeated root, 7.6.1–7.6.66
Indicial polynomial, 7.4.7, 7.5.3
Inductance, 6.3.2
Infinite series, convergent, 12.1.2
Initial-boundary value problem, 12.1.1
- heat equation, 12.1.1–12.1.12
- wave equation, 12.2.1–12.2.68
Initial conditions, 1.2.11
Initial value problems, 1.2.5–1.2.1
- implicit solution of, 2.2.1
- Laplace transforms to solve, 8.3.1–8.3.38
  - formula for, 8.6.2–8.6.10
  - second order equations, 8.3.2–8.3.38
Integral curves, 1.2.1–1.2.3, 8.3.1–1.3.24
Integrals
- convolution, 8.6.5–8.6.6
- improper, 8.1.1
Integrating factors, 2.6.1–2.6.30
- finding, 2.6.6–2.6.30
Interest compounded continuously, 4.1.7–4.1.4
Interval of validity, 1.2.6
Inverse Laplace transforms, 8.1.18–8.2.10
- defined, 8.1.18
- linearity property of, 8.2.1
- of rational functions, 8.2.4–8.2.10
Inverse square law force, motion under, 6.4.10–6.4.1
Irregular singular point, 7.4.1
Isothermal curves, 4.5.16

J

Jump discontinuity, 8.1.12

K

Kepler’s second law, 6.4.1
Kepler’s third law, 6.4.1
Kirchoff’s Law, 6.3.5

L

M

Maclaurin series, 7.1.2
Magnitude of acceleration due to gravity at Earth’s surface, 4.3.2
Malthusian model, 1.1.2
Mathematical models, 1.1.1
- validity of, 4.1.15, 4.2.1, 4.2.17
Matrix/matrices, 10.2.2–10.2.1
- coefficient
  - complex eigenvalue of, 10.6.1–10.6.40
  - defective, 10.5.1
- fundamental, 10.3.2
Mechanics, elementary, 4.3.1–4.4.24
- escape velocity, 4.3.18–4.3.24 4.3.17, 4.3.20
- motion through resisting medium under constant gravitational force, 4.3.3–4.3.17
- Newton’s second law of motion, 4.3.1–4.3.3
- pendulum motion
  - damped, 4.4.5–4.4.1
  - undamped, 4.4.26–4.4.17
- spring-mass system
  - damped, 4.4.26–4.4.5, 6.1.2, 6.2.1–6.2.26
  - undamped, 4.4.8–4.4.26,6.1.1
- units used in, 4.3.1
Midpoint method, 3.2.1
Mixed boundary value problems, 12.3.2
Mixed Fourier cosine series, 11.3.2–11.3.3
Mixed Fourier sine series, 11.3.4
Mixed growth and decay, 4.1.4
Mixing problems, 4.2.3–4.2.1
Models, mathematical, 1.1.1–1.1.3
- validity of, 4.1.15, 4.2.1, 4.2.17
Motion
- damped, 6.1.1
  - critically, 6.2.1
  - overdamped, 6.2.3–6.2.1
  - underdamped, 6.2.2
- elementary, See Mechanics, elementary
- equation of, 6.1.2
- forced, 6.1.3
- free, 6.1.3
- Newton’s second law of, 1.1.10, 4.3.1–4.3.2, 4.4.5, 4.4.9, 4.4.26, 4.4.26–4.4.5, 6.1.1, 6.4.2, 10.1.14
  - autonomous second order equations and, 4.4.6
- simple harmonic, 4.4.12, 6.1.1–6.1.3
  - amplitude of oscillation, 6.1.10
  - frequency of, 6.1.2
  - phase angle of, 6.1.10
- through resisting medium under constant gravitational force, 4.3.1–4.3.17
- under a central force, 6.3.17–6.4.6
- under inverse square law force, 6.4.10–6.4.1
- undamped, 6.1.1
Multiplicity, 9.2.14

N

O

Odd functions, 11.2.2
One-parameter families of curves, 4.5.1–4.5.7
- defined, 4.5.1
- differential equation for, 4.5.2
One-parameter families of functions, 2.1.3
Open interval of convergence, 7.1.2
Open rectangle, 2.3.1
Orbit, 6.4.1
- eccentricity of, 6.4.11
- elliptic, 6.4.11
- period of, 6.4.1
Order of differential equation, 1.2.1
Ordinary differential equation
- defined, 1.2.1
Ordinary point, series solutions of linear second order equations near, 7.2.1–7.3.49
Orthogonality, 11.1.6–11.1.1
- in Sturm-Liouville problems, 13.2.4
Orthogonal trajectories, 4.5.11–4.5.1
- finding, 4.5.9
Orthogonal with respect to a weighting function, 7.2.30, 7.2.31
Oscillation
- amplitude of, 6.1.10
- critically damped, 6.3.12
- overdamped, 6.3.11
- \(RLC\) circuit in forced, with damping, 6.3.17–6.3.2
- \(RLC\) circuit in free, 6.3.8–6.3.17
- undamped forced, 6.1.13–6.1.1
- underdamped, 6.3.9
Oscillatory solutions, 5.4.4–4.4.1, 7.4.21
Overdamped motion, 6.2.1–6.2.3

P

Partial differential equations
- defined, 1.2.1
- Fourier solutions of, 12.1.1–12.4.7
  - heat equation, 12.1.1–12.2.1
  - Laplace’s equation, 12.3.1–12.4.7
  - wave equation 12.2.1–12.2.68
Partial fraction expansions, software packages to find, 8.2.1
Particular solutions of nonhomogeneous higher equations, 9.1.5, 9.3.1–9.4.42
Particular solutions of nonhomogeneous linear second order equations, 5.3.2, 5.3.3–5.3.17, 5.4.1–5.4.7, 5.7.1–5.7.24
Particular solutions of nonhomogeneous linear systems equations, 10.7.1–10.7.23
Pendulum
- damped, 4.4.5–4.4.1
- undamped, 4.4.26–4.4.17
Perigee, 6.4.11
Perihelion distance, 6.4.11
Periodic functions, 8.1.17
Period of orbit, 6.4.1
Phase angle of simple harmonic motion, 6.1.10–6.1.2
Phase plane equivalent, 4.4.4
Piecewise continuous functions, 8.1.4
- forcing, constant coeffocient equations with, 8.5.1–8.5.26
Piecewise smooth function, 11.2.3
- Laplace transforms of 8.1.12–8.1.11, 8.4.4–8.4.36
  - unit step functions and, 8.4.1–8.4.36
Plucked string, wave equation applied to, 12.2.21–12.2.1
Poinccaré, Henri, 4.4.4
Polar coordinates
- central force in terms of, 6.4.1–6.4.1
- in amplitude-phase form, 6.1.10
- Laplace’s equation in, 12.4.1–12.4.7
Polynomial(s)
- characteristic, 5.2.2, 7.3.30, 9.2.2
  - of higher order constant coefficient homogeneous equations, 9.2.2–9.2.9
- Chebyshev, 7.2.16
- indicial, 7.4.7, 7.5.3
- Taylor, 7.1.6
- trigonometric, 11.2.30
Polynomial operator, 9.2.2
Population growth and decay, 1.1.1
Positive half-plane, 10.5.21
Potential equation, 12.3.2
Power series, 7.1.1–7.1.25
- convergent, 7.1.1–7.1.1
- defined, 7.1.1
- differentiation of, 7.1.4–7.1.9
- divergent, 7.1.1
- linear combinations of, 7.1.5–7.1.1
- radius of convergence of, 7.1.2, 7.1.3
- shifting summation index in, 7.1.10–7.1.4
- solutions of linear second order equations, represented by, 7.2.1–7.3.49
- Taylor polynomials, 7.1.5
- Taylor series, 7.1.2
- uniqueness of 7.1.5–7.1.6

Q

Quasi-period, 6.2.3

R

Radioactive decay, 4.1.2–4.1.7
Radius of convergence of power series, 7.1.2, 7.1.3
Rational functions, inverse Laplace transforms of, 8.2.4–8.2.10
Rayleigh, Lord, 4.4.20
Rayleigh’s equation, 4.4.20
Real part, 5.2.17
Rectangle, open, 2.3.1
Rectangular grid, 1.3.1
Recurrence relations, 7.2.16
- in Frobenius solutions, 7.5.14
- two term, 7.5.20–7.5.26
Reduction of order, 5.2.10, 5.6.1–5.6.40
Regular singular points, 7.4.1–7.4.23
- at \(x_{0}=0\), 7.5.1–7.5.68
Removable discontinuity, 8.1.12
Resistance, 6.3.2
Resistor, 6.3.2
Resonance, 6.1.1
Ricatti, Jacopo Francesco, 2.4.55
Ricatti equation, 2.4.55
\(RLC\) circuit, 6.2.26–6.3.1
- closed, 6.2.26
- in forced oscillation with danping, 6.3.17
- in free oscillation, 6.3.8–6.3.17
Roundoff errors, 3.1.2
Runge-Kutta method, 3.1.7, 3.3.1–3.3.28, 10.1.1
- for cases where \(x_{0}\) isn’t the left endpoint, 3.3.3
- for linear systems of differential equations, 10.1.1
- semilinear, 3.1.1, 3.3.4

S

T

Tangent lines, 4.5.6
Taylor polynomials, 7.1.5
Taylor Series, 7.1.2
Temperature, Newton’s law of cooling, 1.1.5 4.2.1–4.2.2, 4.2.1–4.2.17
Temperature decay constant of the medium, 4.2.1
Terminal velocity, 4.3.1
Time-varying amplitude, 6.2.3
Total impulse, 8.6.14
Trajectory(ies)
- of autonomous second order equations, 4.4.4
- orthogonal, 4.5.11–4.5.1
  - finding, 4.5.9–5.5.1
- shadow, 10.6.13
of \(2\times 2\) systems, 10.4.18–10.4.1, 10.5.19–10.5.1, 10.6.10–10.6.1
Transfer functions, 8.6.3
Transformation of nonlinear equations to separable first order, equations, 2.4.1–2.5.29
- Bernoulli’s equation, 2.4.2
- homogeneous nonlinear equations, 2.4.7–2.4.1
- other equations, 2.4.4
Transform pair, 8.1.1
Transient current, 6.3.2
Transient components, 6.2.18, 8.6.21
Transient solutions, 6.3.1
Trapezoid rule, 3.2.32
Trivial solution
- of homogeneous linear first order equations, 2.1.1
- of homogeneous linear second order equations, 5.1.1
- of homogeneous linear systems of differential equations, 10.3.1
- of linear higher order differential equations, 9.1.1
Truncation error(s), 3.1.2
- in Euler’s method, 3.1.9
- global, 3.1.18, 3.2.1
- local, 3.1.8
  - numerical methods with \(O(h^{3})\), 3.2.7–3.2.1
Two-point boundary value problems, 13.1.1–13.1.32
- assumptions, 13.1.5
- boundary conditions for, 13.1.5
- defined, 13.1.5
- Green’s function for, 13.1.19
- homogeneous and nonhomogeneous, 13.1.1

U

Undamped autonomous second order equations, 4.4.7–4.4.23
- pendulum, 4.4.26–4.4.17
- spring-mass system, 4.4.8–4.4.12
- stability and instabilty conditions for, 4.4.18–4.4.23
Undamped motion, 6.1.1
Underdamped motion, 6.2.2
Underdamped oscillation, 6.3.9
Undetermined coefficients
- for linear higher order equations, 9.3.1–9.3.75
  - forcing functions, 9.3.1–9.3.1
- for linear second order equations, 5.4.1–5.5.40
- principle of superposition, 5.4.7
Uniqueness of solutions of nonlinear first equations, 2.3.1–2.3.21
Uniqueness theorem, 2.1.2, 2.3.1, 5.1.1, 9.1.1, 10.2.1
Unit step function, 8.4.1–8.4.36

V

Validity, interval of, 1.2.6
Vandermonde, 9.2.39
Vandermonde determinant, 9.2.39
van der Pol’s equation, 4.4.19
Variables, separation of, 2.1.16, 2.2.1
Variation of parameters
- for linear first order equations, 2.1.16
- for linear higher order equations, 9.4.1–9.4.42
  - derivation of method, 9.4.1–9.4.5
  - fourth order equations, 9.4.2–9.4.1
  - third order equations, 9.4.7
- for linear higher second order equations, 5.7.1
- for nonhomogeneous linear systems of differential equations, 10.7.1–10.7.23
Velocity
- escape, 4.3.18–4.3.1
- terminal, 4.3.1–4.3.4
Verhulst, Pierre, 1.1.4
Verhulst model, 1.1.4, 1.3.24, 2.4.14
Vibrating strings, wave equation applied to, 12.2.1
Vibrations
- forced, 6.2.11–6.2.1
- free, 6.2.1–6.2.11
Voltage, impressed, 6.2.26
Voltage drop, 6.3.1
Volterra, Vito 8.6.11
Volterra integral equation, 8.6.11

W

Wave equation, 12.2.1–12.2.68
- defined, 12.2.1
- plucked string, 12.2.19–12.2.1
- vibrating string, 12.2.1–12.2.1
  - assumptions, 12.2.1
  - formal solution, 12.2.5–12.2.1
Wave, traveling, 12.2.23
- orthogonal with respect to, 11.2.7 7.2.30
Wronskian
- of solutions of homogeneous linear systems of differential equations, 10.3.2
- of solutions of homogeneous second differential equations, 5.1.21–5.1.5
- of solutions of homogeneous linear higher order differential equations, 9.1.13–9.1.5