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As FirstOrder System
Many numerical timeintegration techniques are usually applied to first order ODE systems in the literature. It is simple enough to expand the definition of the simple harmonic oscillator, from (3.1), to a first order system in two variables, i.e., in matrix form,

(3.15) 
All frequency domain and energy analysis of course applied equally to this equivalent system. Though firstorder systems will only rarely appear in this book, it is worth being aware of such systems, as many timeintegration techniques, including the ubiquitous RungeKutta family of methods, are indeed usually presented with reference to first order systems. In the distributed case, the important ``finitedifference time domain" family of methods for hyperbolic PDEs[215], such as the defining equations of electromagnetics, are also usually applied to first order systems.
Stefan Bilbao
20061115