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Higherorder ordinary differential equations
CONTENTS 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 The Fourier coefficients Symmetry considerations Discontinuous functions Non-periodic functions Integration and differentiation Complex Fourier series Parseval’s theorem Exercises Hints and answers 417 419 420 422 424 424 426 427 431 13 13.1 Integral transforms Fourier transforms 433 433 The uncertainty principle; Fraunhofer diffraction; the Dirac δ-function; relation of the δ-function to Fourier transforms; properties of Fourier transforms; odd and even functions; convolution and deconvolution; correlation functions and energy spectra; Parseval’s theorem; Fourier transforms in higher dimensions 13.2 Laplace transforms 453 Laplace transforms of derivatives and integrals; other properties of Laplace transforms 13.3 13.4 13.5 Concluding remarks Exercises Hints and answers 459 460 466 14 14.1 14.2 First-order ordinary differential equations General form of solution First-degree first-order equations 468 469 470 Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; isobaric equations; Bernoulli’s equation; miscellaneous equations 14.3 Higher-degree first-order equations 480 Equations soluble for p; for x; for y; Clairaut’s equation 14.4 14.5 Exercises Hints and answers 484 488 15 15.1 Higher-order ordinary differential equations Linear equations with constant coefficients 490 492 Finding the complementary function yc (x); finding the particular integral yp (x); constructing the general solution yc (x) + yp (x); linear recurrence relations; Laplace transform method 15.2 Linear equations with variable coefficients The Legendre and Euler linear equations; exact equations; partially known complementary function; variation of parameters; Green’s functions; canonical form for second-order equations x 503