## Abstract

Simple mathematical solutions can be found for linear ordinary differential equations having constant coefficients. Such equations form the basis of many well-known and very useful lumped parameter models. Introductory courses in dynamics and in electrical circuit theory provide many illustrations of models of this type. For example, transients in simple electrical circuits involving linear elements such as ideal resistors, capacitors and inductors can be studied very easily using standard analytical tools for the solution of ordinary differential equations. Concepts such as the complementary function, particular integral, D-operator and Laplace transform are of considerable importance for problems of this type, but such analytical tools are of no use in the case of many practical problems involving nonlinear models. For example, analytical methods are of little assistance if an inductive element in an electrical circuit has a core of magnetic material and displays significant hysteresis. Similarly, in dynamics, linear mechanical systems involving elements with mass, spring stiffness and viscous resistance provide a basis for much elegant mathematical analysis which can provide valuable understanding. Replacement of a viscous damping element by an element with static friction immediately makes the problem nonlinear and eliminates the possibility of applying standard mathematical techniques. In cases where analytical methods are impractical the only possible approach is through numerical techniques of some kind or by the use of an analog method.

### Keywords

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