Appendix: Linear Algebra and Solving Difference Equations and Systems of Differential Equations
This appendix contains basic notions from undergraduate mathematics. This appendix is no replacement for a Linear Algebra book or a Differential Equations class but we will state the important concepts and results that are used throughout this book. Furthermore, I am going to assume that the reader is familiar with basic notions of matrix algebra such as determinant, rank of a matrix, linearly independent vectors, inverse of a matrix, when such inverse exists, etc. Such notions should be taught in high school. For more details we refer to Hoffman and Kunze (1971) for linear algebra and Nagle et al. (2011) for differential equations and DiPrima (2004) for Ordinary Differential Equations also containing the theory on difference equations.
A.1 Solving difference equations with constant coefficients
This methodology is given for second order difference equations but higher order equations are solved in a very similar way. Suppose we are given an equation of the form:
with some boundary conditions.
The idea is to look for solutions of the form , with c some constant and y needs to be determined. Note that if we have two solutions of this form (say and ), then any linear combination ...