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Up: The Simple Harmonic Oscillator
Many numerical time-integration 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,
As First-Order System
All frequency domain and energy analysis of course applied equally to this equivalent system. Though first-order systems will only rarely appear in this book, it is worth being aware of such systems, as many time-integration techniques, including the ubiquitous Runge-Kutta family of methods, are indeed usually presented with reference to first order systems. In the distributed case, the important ``finite-difference time domain" family of methods for hyperbolic PDEs, such as the defining equations of electromagnetics, are also usually applied to first order systems.