Second Course in Ordinary Differential Equations for Scientists and Engineers

O: Review.- 1. Solution of second order ordinary differential equations by series.- 2. Regular singular points.- 3. Series solutions near a regular singular point.- 1: Boundary Value Problems.- 1. Introduction.- 2. Adjoint differential equations and boundary conditions.- 3. Self -adjoint systems.- 4. A broader approach to self-adjoint systems.- 5. Sturm-Liouvi1 le theory.- 6. Introduction to orthogonality and completeness.- 2: Special Functions.- 1. Hypergeometric series.- 2. Bessel functions.- 3. Legendre polynomials.- 4. Gamma function.- 3: Systems of Ordinary Differential Equations.- 1. Introduction.- 2. Method of elimination.- 3. Some linear algebra.- 4. Linear systems with constant coefficients.- 5. Linear systems with variable coefficients.- 6. Elements of linear control theory.- 7. The Laplace transform.- 4: Applications of Symmetry Principles to Differential Equations.- 1. Introduction.- 2. Lie groups.- 3. Lie algebras.- 4. Prolongation of the action.- 5. Invariant differential equations.- 6. The factor ization method.- 7. Examples of factorizable equations.- 5: Equations with Periodic Coefficients.- 1. Introduction.- 2. Floquet theory for periodic equations.- 3. Hill’s and Mathieu equations.- 6: Green’s Functions.- 1. Introduction.- 2. General definition of Green’s function.- 3. The interpretation of Green’s functions.- 4. Generalized functions.- 5. Elementary solutions and Green’s functions.- 6. Eigenfunetion representation of Green’s functions.- 7. Integral equations.- 7: Perturbation Theory.- 1. Preliminaries.- 2. Some basic ideas-regular perturbations.- 3. Singular perturbations.- 4. Boundary layers.- 5. Other perturbation methods.- *6. Perturbations and partial differential equations.- *7. Perturbation of eigenvalue problems.- *8. The Zeeman and Stark effects.- 8: Phase Diagrams and Stability.- 1. General introduction.- 2. Systems of two equations.- 3. Some general theory.- 4. Almost linear systems.- 5. Almost linear systems in R2.- 6. Liapounov direct method.- 7. Periodic solutions (limit cycles).- 9: Catastrophes and Bifurcations.- 1. Catastrophes and structural stability.- 2. Classification of catastrophe sets.- 3. Some examples of bifurcations.- 4. Bifurcation of equilibrium states in one dimension.- 5. Hopf bifurcation.- 6. Bifurcations in R.- 10: Sturmian Theory.- 1. Some mathematical preliminaries.- 2. Sturmian theory for first order equations.- 3. Sturmian theory for second order equations.- 4. Prufer transformations.