There are two main objectives of this text. The first is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. The second is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations.
Ordinary differential equations and their solutions: Classification of differential equations, Solutions, Implicit solutions, Singular solutions, Initial Value Problems, Boundary Value Problems, Basic existence and uniqueness theorems (statement and illustration only), Solution of first order equations: Separable equations, Linear equations, Exact equations, Special integrating factors, Substitutions and transformations, Modeling with first order differential equations, Model solutions and interpretation of results, Orthogonal and oblique trajectories, Solution of higher order linear equations: Linear differential operators, Basic theory of linear differential equations, Solution space of homogeneous linear equations, Fundamental solutions of homogeneous solutions, Reduction of orders, Homogeneous linear equations with constant coefficients, Non homogeneous equation, Method of undetermined coefficients, Variation of parameters, Euler Cauchy differential equation, Modeling with second order equations, Initial Value Problems and Boundary Value problems, reduction of order, Euler equation, generating functions, eigenvalue problems. Inhomogeneous linear difference equations (variation of parameters, reduction of order, Series solutions of second order linear equations: Taylor series solutions about an ordinary point. Frobenious series solutions about regular singular points. Series solutions of Legendre, Bessel, Laguerre and Hermite equations. Systems of linear first order differential equations: Elimination method. Matrix method for homogeneous linear systems with constant coefficients. Variation of parameters. Matrix exponential. 2b1af7f3a8