Index

erf(x), 141 capacitor, 42 carrying capacity, 260 Euler-Mascheroni constant, 150 Cauchy-Euler equations, 65, 143 nonhomogeneous, 68 air drag, 114 center, 219 air resistance, 15 Chain Rule, 319 amplitude, 41 characteristic equation, 34, 65, 237 analytic, 145 chemical kinetics, 231 angular frequency, 41 chili problem, 30 anonymous function, 82 Clairaut equation, 23 associated Legendre functions, 161 Clairaut’s Theorem, 8 atmospheric density, 105 Clairaut, Alexis Claude, 23 autonomous, 213, 264 classical orthogonal polynomials, 161, 162 coefficient matrix, 235 Baumgartner, Felix, 104 conservation beat frequency, 54 angular momentum, 118 Bernoulli equation, 22, 262 energy, 120 Bernoulli, Daniel, 22, 168 constant coefficient equations, 34 Bernoulli, Jacob, 22 complex roots, 36 Bernoulli, Jacob II, 22 repeated roots, 35 Bernoulli, Johann, 22 convolution Bernoulli, Johann II, 22 Laplace transform, 199 Bernoulli, Johann III, 22 convolution theorem Bernoulli, Nicolaus I, 22 Laplace transform, 198 Bernoulli, Nicolaus II, 22 cooling, 14 Bessel equation, 151 cosmological constant, 124 Bessel functions, 168 coupled systems, 213 first kind, 169 Cramer’s rule, 203, 216 generating function, 171 creeping flow, 131 identities, 170 critical density, 125 order one half, 155 current, 42 orthogonality, 171 curvature tensor, 124 recursion formula, 171 cycloid, 127 second kind, 170 Bessel, Friedrich Wilhelm, 169 bifurcation, 266 damped harmonic motion, 220 bifurcation point, 266 de Moivre’s Formula, 343 Big Chill, 129 de Moivre, Abraham, 343 Big Crunch, 129 density parameter, 125 big-Oh, 348 DEplot, 89 binomial coefficients, 344 derivatives, 311 binomial expansion, 343 table, 319 356 differential equations

DEtools, 89 Euler’s Method, 75, 120 difference equation, 185 Euler, Leonhard, 22, 168, 298 differential equations, 188 Euler-Cromer method, 102 autonomous, 5 Exact differential equations, 8 first order, 4, 6 expanding universe, 123 linear, 4, 31 exponential function, E1(x), 150 nonhomogeneous, 49 exponential of a matrix, 235 Runge-Kutta methods, 95 ezplot, 80 second order, 31 separable, 5 Fagnano, Giulio Carlo, 298 series, 169 Feynman’s trick, 327 Taylor methods, 90 Feynman, Richard, 181, 317, 327 differential operator, 31 Fibonacci sequence, 344 Differentiation Under Integral, 326 fixed point, 261, 264 Dirac delta function force Laplace transform, 196 drag, 108, 114 sifting property, 195 lift, 108 Dirac, Paul Adrien Maurice, 194 Magnus, 108 direction field, 84, 218 Fourier, Joseph, 1, 168 direction fields, 81 free fall, 1 distribution, 194 frequency, 41 double factorial, 165 Friedmann’s Equation, 124 double tank problem, 230 Friedmann, Alexander Alexandrovich, drag coefficient, 130 124 drag force, 15, 105, 108, 114 Frobenius method, 147 dsolve, 80, 88 function Duffing equation, 291 average, 329 Duffing, Georg, 291 sigmoid, 264 functions exponential, 312 eigenvalue, 237 hyperbolic, 317 eigenvalue problem, 237 inverse trigonometric functions, 317 eigenvector, 237 logarithmic, 312 Einstein field equation, 124 polynomial, 312 Einstein tensor, 124 rational, 312 Einstein, Albert, ix, 123, 211 trigonometric, 313 Einstein-de Sitter universe, 126 fundamental matrix, 245 ellipse, 119 fundamental matrix solution, 246 elliptic functions, 301 Fundamental Theorem of Calculus, 2, elliptic integral, 297, 298 321, 327 complete, 299 incomplete, 299, 300 epidemic model, 233 Gödel, Kurt, 31 equilibrium, 215 Gamma function, 171, 187 center, 219 generalized function, 194 degenerate node, 221 geometric series, 334 focus, 220 GNU Octave, 85 node, 219 gravitational potential, 163 saddle, 219 Green’s functions, 32 source, 219 initial value, 60 equilibrium solution, 261, 264 Green, George, 61 error function, 141, 176 Gregory, James, 336 Escher, M. C., 75 Gudermann, Christoph, 323 Euler’s Formula, 342 Gudermannian, 323 index 357

half-life, 13 Point rule, 204, 228 Hankel, Hermann, 135 Kittinger, Joe, 104 harmonic series partial sum, 161 Lagrange equation, 23 Hawking, Stephen, 129 Lagrange, Joseph Louis, 23 Heaviside function, 191 Lagrange, Joseph-Louis, 168 heteroclinic orbits, 279 Lambert, Johann Heinrich, 323 hold command, 82 laminar flow, 130 homogeneous, 213 Laplace transform, 181 Hooke’s Law, 40 convolution, 198 Hubble’s constant, 125 differential equations, 188 Hubble, Edwin, 124 periodic functions, 193 Huen’s method, 97 properties, 187 hyperbolic cosine, 317 transform pairs, 183 hyperbolic function Laplace, Pierre-Simon, 181 identities, 318 Law of cooling, 14 substitution, 331 LC circuit, 46 hyperbolic sine, 317 Legendre equation, 142 hyperbolic tangent, 318 Legendre polynomials, 161 generating function, 163 identities normalization, 167 double angle, 314 recurrence relation, 162 half angle, 314 recursion formula, 162 product, 316 Rodrigues Formula, 162 Pythagorean, 313 Legendre, Adrien-Marie, 161, 171 sum and difference, 314 Lemaître, Georges, 124 tangent, 313 lift force, 108 implicit solution, 5 line of equilibria, 222 Implicit-Euler Method, 121 linear operator, 31 impulse function linearity, 31 unit impulse, 196 linearization, 264, 271 indicial equation, 148 linearly dependent, 32 inductor, 42 logarithmic differentiation, 157 initial value problem, 4, 188 logistic equation, 260 integral transforms, 181 solution, 263 integrals, 311 Lotka, Alfred James, 281 integration limits, 322 Lotka-Volterra model, 281 simple substitution, 321 first integral, 284 table, 320 LRC circuits, 42, 227 trigonometric, 328 lsode, 85 integrating factor, 6 integration by parts, 323 integration by recursion, 185 Maclaurin series, 135, 235, 338 interval of convergence, 340 Maclaurin, Colin, 336 inverse Laplace transform, 189 Magnus force, 108 Malthus, Thomas Robert, 12, 260 Jacobian matrix, 270 Maple, 88 mass-spring system, 40 Kepler’s Laws of Planetary Motion, 119 coupled, 212 Kepler, Johannes, 168 MATLAB, 79 kernel, 61 matrix formulation, 234 kinematic viscosity, 105, 130 mechanical energy, 297 Kirchoff’s rules, 42 megaparsec, 124 Loop rule, 204, 228 Method of Reduction of Order, 37, 141, 358 differential equations

150 pitchfork bifurcation, 268 Method of undetermined coefficients, 32, Pochhammer symbol, 174 49 Pochhammer, Leo August, 174 modified, 52 Poincaré surface of section, 292 Midpoint method, 96 Poincaré, Jules Henri, 259 mixture problems, 17, 229 point at infinity, 177 modified Euler method, 97 Poisson, Siméon, 168 polar coordinates, 223 n-body problem, 117 derivatives, 223 nested radicals, 315 population models, 260 Neumann function, 170 logistic, 260 Newton’s Law of Cooling, 14 Lotka-Volterra, 281 Newton’s Law of Gravitation, 4, 117 Malthusian, 11, 260 Newton’s second law, 1, 40 potential drops, 42 Newton, Isaac, 14 potential energy, 297 nonautonomous, 213 predator-prey model, 232 nonlinear systems, 290 principal matrix solution, 248 nonhomogeneous solutions, 249 pursuit curves, 19 nonlinear pendulum Python, 86 forced, 290 numerical, 100 quadrature, 297 period, 296 quiver, 82, 84 numerical solutions, 75 radicals ode45, 80, 98 nested, 315 second order, 83 radioactive decay, 12 odeint, 86 radius of convergence, 340 one-form, 8 RC circuits, 43 orbit, 217 re-indexing series, 138 ordinary differential equation, 4 Reduction of Order, 35, 37, 141, 146 ordinary point, 145 relative error, 296 orthogonal polynomials, 162 resistor, 42 oscillations resonance, 54 coupled, 211 Reynolds number, 105, 115, 130 damped, 47 Reynolds, Osborne, 130 forced, 52 Riccati equation, 26, 262 underdamped, 48 Riccati, Jacopo Francesco, 26, 262 overdamped, 221 Riemann, Georg Friedrich Bernhard, 61 RMS voltage, 329 partial fraction decomposition, 16, 189 Romeo and Juliet, 233 particular solution, 5 root mean square, 329 Pascal’s triangle, 344 Runge-Kutta, 95 Pascal, Blaise, 344 fourth order, 97 pendulum, 40 second order, 96 forced nonlinear, 290 Runge-Kutta-Fehlberg method, 98 nonlinear, 269 Russell, Bertrand, 311 period, 296 pendulum, simple, 41 saddle point, 219 period, 41 saddle-node bifurcation, 267 periodic function, 317 salsa problem, 257 periodic functions scale factor, 124 Laplace transform, 193 scheme phase plot, 84 Laplace transform, 181 phase portrait, 216 Schwartz, Laurent, 194 index 359

scipy, 86 symplectic integrators, 121 semi-implicit Euler Method, 102 systems separatrix, 279 constant coefficient, 213 sequence Fibonacci, 344 Tabular Method, 325 series tank problem, 17 binomial series, 346 Taylor polynomials, 96, 337 geometric series, 334 Taylor series, 135, 337 Maclaurin series, 135, 338 Taylor, Brook, 336 power series, 135, 336 terminal velocity, 15 re-indexing, 138 The indicial equation., 149 Taylor series, 135, 337 time constant, 44 telescoping, 335, 352 time dilation, 346 sigmoid function, 264 trajectory, 217 simple harmonic motion, 40, 196, 212, transcritical bifurcation, 267 221 transforms simple harmonic oscillator, 83 Laplace, 181 singular point, 145 transitional flow, 131 singular solutions, 24 trigonometric sky diving, 104 identities, 313 slope field, 217 substitution, 329 small angle approximation, 42 turbulent flow, 131 soccer ball, 107 solitons, 317 underdamped, 221 solutions unstable node, 219 equilibrium, 215 Special Relativity, 347 van der Pol, Balthasar, 285 spiral, 220, 223 variation of parameters, 32, 57 sports balls, 107 vector spaces, 31 square wave, 193 Velocity Verlet Method, 122 Störmer, Fredrik Carl Mülertz, 122 Verhulst, Pierre François, 260 Störmer-Verlet Method, 122 Verlet Method, 122 stability criteria, 265 Verlet, Loup, 122 step function, 191 viscosity, 130 Stokes drag force, 131 Volterra integral equation, 208 Sturm-Liouville, 171 Volterra, Vito, 208 substitution von Ettingshausen, Andreas Freiherr, 345 hyperbolic, 331 trigonometric, 329 Watson, G.N., 168 surfaces of section, 292 Wronskian, 33, 247