## Chapter 2

This chapter investigates the fundamental theory and the essential techniques employed in the study of linear initial value problems. An inherent property which makes linear systems simple to deal with, is the superposition principle. We begin this chapter with this principle and discuss some of its consequences. Then, we collec:-t several definitions and results from algebra whieh arc used in later chapters also. Next, the <:oncept of linearly independent functions, the Casoratian matrix, the fundamental matrix solution, and its explicit representation along with its properties arc discus..<ied iu detail. Here an interesting example from Markov chains is also illustrated. This is followed by the method of variation of constants for the solutions of nonhomogeneous difference systems. Next we discuss adjoint systems and develop adjoint identities which arc used in Chapter 8. Then, we consider the systems with constant coefficients and provide some constructive methods for their closed form solutions. These methods do not use Jordan form and <"an easily be mastered. A very important aspect of the qualitative study of the solutions of difference systems is their periodicity. In Sections 2.9 and 2.10 respectively, we provide necessary and sufficient conditions so that the solutions of a given system are periodic and almost periodic. Eventhough, higher order equations are expressible as difference systems, they merit some special attention. In Section 2.11 we incorporate the method of variation of constants, the concepts of exact and adjoint equations, and Lagrange's and Green's identities. This is followed by the method of generating functions, which is a very elegant technique for obtaining the closed form solutions of higher order difference equations. Higher order difference equations with constant coefficients find an application in computing the roots of a given polynomial. This classical method originally due to Bernoulli is presented in Section 2.13. Bernoulli's method also provides the motivation of several important results, e.g. Poincare's and Perron's theorems, which we discuss in Section 2.14. In Section 2.15 we introduce and illustrate the regular and singular perturbation techniques for the construction of the solutions of difference equations.