Historical Perspective

  • J. A. Richards
Part of the Communications and Control Engineering Series book series (CCE)


The field of linear differential equations with constant coefficients has been extensively studied as a unified body of knowledge; standard forms of solution are well-known and Laplace transform techniques can be readily applied to obtain both natural and forced responses. Consequently a large proportion of all physical systems, including a majority of electrical network configurations, can be adequately described mathematically. However when the constraints of linearity and constant coefficients are relaxed, the neatness of solution is lost and very often particular non-linear and/or time-varying1 systems and their associated differential equations have to be treated individually. Techniques devised for one type of system often cannot be generalised for use with another and consequently little or nothing is gained by developing stylised solution methods for the equations, such as those based upon integral transforms. Indeed it can even be difficult delineating classes of equation in many instances. A case which is an exception however is that of linear differential equations with coefficients that are periodically varying with time. As a class, so-called periodic differential equations exhibit similarities in behaviour, even though the solutions in most cases are not known in closed form, a feature which is exploited in Chapter five in developing modelling techniques for describing the dynamic behaviour of periodically varying systems.


Linear Differential Equation Parametric Excitation Hill Equation Capacitor Voltage Alfven Wave 
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Copyright information

© Springer-Verlag Berlin, Heidelberg 1983

Authors and Affiliations

  • J. A. Richards
    • 1
  1. 1.School of Electrical Engineering and Computer ScienceUniversity of New South WalesKensingtonAustralia

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