Some general concepts: Definition and classification. Differential equations in Physics. Solutions of differential equations: Existence, uniqueness, and methods.
First order equations. Definition and geometrical understanding. Exact equations: equations in separated variables. Integrating factors: separable and linear equations. Transformation methods: homogeneous equations, Bernoulli, Ricatti. .
Implicit equations for the derivative: Clairut and Lagrange..
Higher order equations. Definition and geometrical understanding. Order reduction. Functions linear dependence. Homogeneous linear equations: fundamental system of solutions and Liouville's formula. Complete linear equations:
variation of constants and Cauchy's method. Generalized functions and fundamental solution.
Homogeneous linear equations with constant coefficients: characteristic equation. Complete linear equations with constant coefficients: annulling operator and inverse operator. Cauchy-Euler equations.
Systems of ordinary differential equations. Definition and geometrical understanding. Reduction to one equation. First integrals. First order homogeneous linear systems: fundamental system of solutions. Complete first order linear systems: variation of constants , Cauchy's method. First order linear systems with constant coefficients.
Laplace transform Definition and properties. The inverse transform. Convolution. Application to initial value problems for constant coefficient linear equations and systems.
Series solutions of linear differential equations.
Ordinary and singular regular points. Frobenius' method. Applications: special functions and associated equations.
Approximation methods for ordinary differential equations.Graphical methods. Power series in initial value probles: Taylor series and indeterminate coefficients. Picard's method of successive approximations. Perturbation theory. Numerical methods: elementary, one step (Runge-Kutta), multistep (predictor-corrector) and extrapolation (Bulirsch-Stoer). Problems and caveats.
Nonlinear equations and stability theory. The concept
of stability. Fixed points. Stability of linear systems. Linear stability. Conservative systems. Lyapunov functions.
Limit cycles: Poincaré - Bendixson's theorem. An introduction to strange attractors and deterministic chaos.
Fundamental theory of ordinary differential equations. Theorem of existence and uniqueness. Dependence on initial conditions; dependence on a parameter.
BIBLIOGRAPHY
Texts
J. M. Aguirregabiria Ecuaciones diferenciales ordinarias
para estudiantes de fsica UPV/EHU (2000)
W. E. Boyce y R. C. DiPrima Ecuaciones diferenciales y
problemas con valores en la frontera 4a Ed., Limusa (1998)
K. F. Riley, M. P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering Cambridge University Press (2nd ed. 2002))
M. D. Greenberg Foundations of applied mathematics Prentice-Hall (1978)
J. Mathews and R.L. Walker Mathematical methods of physics Benjamin (1970)
Problems (in spanish)
F. Ayres Ecuaciones diferenciales Schaum McGraw-Hill(1991)
A. I. Kiseliov, G. I. Makarenko y M. L. Krasnov y Problemas
de ecuaciones diferenciales ordinarias, 9a Ed.,Mir-Rubios
1860 (1992)
M. L. Krasnov, A. I. Kiseliov y G. I. Makarenko Funciones
de variable compleja. Cálculo operacional. Teoría de la estabilidad.
Mir-Rubios
1860 (1992)
Tables (only the spanish versions are mentioned, but english versions do exist)
M. R. Spiegel y L. Abellanas Fórmulas y Tablas
de Matemática Aplicada Schaum McGraw-Hill (1999)
I. Bronshtein y K. Semendiaev Manual de Matemáticas
Mir (1993). Handbook of Mathematics Springer (1997)
