Differential and Difference Equations

In this chapter we treat various questions about ordinary differential and difference equations. We first define and give various properties of the exponential of a matrix, which allows us to express the solution of a system of differential equations in a concise way. The Jordan form is the main tool that allows us to obtain the basic properties of a matrix exponential, and through these properties we are able to express the solution of a system of differential equations ẋ = Ax with constant coefficients in terms of the eigensystem of A. Higher-order equations can be reduced to a first-order system and, thus, treated in the same way. In Section 5.2 we ascertain the stability of solutions when the initial condition is changed, and in Section 5.3 we obtain corresponding results for difference equations. These stability results for difference equations can be interpreted also as convergence theorems for certain iterative methods. Finally, in Section 5.4, we treat Lyapunov’s criterion for stability as well as several related results.