Nonlinear and Nonconstant Equations

The previous first and second order differential equations were extremely simple. They had derivatives that were raised only to the first power, they had constant coefficients, and they had only one independent variable, time. Not aU equations used to describe physical or biological phenomena are so simple. 

Many biological happenings are nonlinear. They may oscillate, but not with any set frequency. They may form exponential-like responses, but cannot be characterized by one time constant. Input-output relationships may not follow idealized forms. In these cases, the biological engineer must either resort to nonlinear equations or to numerical solutions to describe these phenomena. 

A case in point is adaptive control systems, to be described in the next section. Most control systems are based upon first and second order differential equations, but adaptive systems can change their responses to satisfy special requirements. Biological systems are particularly adept at this they can often change the type or magnitude of response when simple predetermined responses are 

Choose an equation that adequately matches the essence of the response. 

Incorporate the equation in a model that can be used to predict future responses. 

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