Mathematical Methods for Physics and Engineering - Matematica.NET

CONTENTS15.3 General ordinary differential equations 518Dependent variable absent; independent variable absent; non-linear exactequations; isobaric or homogeneous equations; equations homogeneous in xor y alone; equations having y = Ae x as a solution15.4 Exercises 52315.5 Hints and answers 52916 Series solutions of ordinary differential equations 53116.1 Second-order linear ordinary differential equations 531Ordinary and singular points16.2 Series solutions about an ordinary point 53516.3 Series solutions about a regular singular point 538Distinct roots not differing by an integer; repeated root of the indicialequation; distinct roots differing by an integer16.4 Obtaining a second solution 544The Wronskian method; the derivative method; series form of the secondsolution16.5 Polynomial solutions 54816.6 Exercises 55016.7 Hints and answers 55317 Eigenfunction methods for differential equations 55417.1 Sets of functions 556Some useful inequalities17.2 Adjoint, self-adjoint and Hermitian operators 55917.3 Properties of Hermitian operators 561Reality of the eigenvalues; orthogonality of the eigenfunctions; constructionof real eigenfunctions17.4 Sturm–Liouville equations 564Valid boundary conditions; putting an equation into Sturm–Liouville form17.5 Superposition of eigenfunctions: Green’s functions 56917.6 A useful generalisation 57217.7 Exercises 57317.8 Hints and answers 57618 Special functions 57718.1 Legendre functions 577General solution for integer l; properties of Legendre polynomials18.2 Associated Legendre functions 58718.3 Spherical harmonics 59318.4 Chebyshev functions 59518.5 Bessel functions 602General solution for non-integer ν; general solution for integer ν; propertiesof Bessel functionsxi