Secondorder ordinary differential equations
Special functions, SturmLiouville theory and transforms
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About the book
Description
Ordinary differential equations, and secondorder equations in particular, are at the heart of many mathematical descriptions of physical systems, as used by engineers, physicists and applied mathematicians. This text provides an introduction to all the relevant material normally encountered at university level: series solution, special functions (Bessel, etc.), SturmLiouville theory (involving the appearance of eigenvalues and eigenfunctions) and the definition, properties and use of various integral transforms (Fourier, Laplace, etc.). Numerous worked examples are provided throughout.
Content
 Series solution: essential ideas
 ODEs with regular singular points
 Exercises 1
 The basic method
 The two special cases
 Exercises
 First solution
 The second solution
 The modified Bessel equation
 Exercises 4
 Exercises 5
 Legendre polynomials
 Hermite polynomials
 Bessel functions
 Exercises 6
 The secondorder equations
 The boundaryvalue problem
 Selfadjoint equations
 Exercises 1
 Real eigenvalues
 Simple eigenvalues
 Ordered eigenvalues
 Exercises 2
 The fundamental oscillation theorem
 Using the fundamental oscillation theorem
 Orthogonality
 Eigenfunction expansions
 Exercises 3
 Exercise 4
 The appearance of an integral transform from a PDE
 The appearance of an integral transform from an ODE
 Exercise 1
 LTs of some elementary functions
 Some properties of the LT
 Inversion of the Laplace Transform
 Applications to the solution of differential and integral equations
 Exercises 2
 FTs of some elementary functions
 Some properties of the FT
 Inversion of the Fourier Transform
 Applications to the solution of differential and integral equations
 Exercises 3
 HTs of some elementary functions
 Some properties of the HT
 Application to the solution of a PDE
 Exercises 4
 MTs of some elementary functions
 Some properties of the MT
 Applications to the solution of a PDE
 Exercises 5