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Calculus Primer 9: Nature's Math
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Copyright © 2004, P. Lutus

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We now turn to an interesting differential equation that has many applications. Its simplicity of expression belies its wide application in various branches of physics. It has several versions, we will examine two.


Voltage at capacitor in series RC circuit.
Version 1:

(1)  y(t) + r c y'(t) = b
(2)  y(0) = a

The solution for these terms is:

(3)  y(t) = b + (a-b) e-t/rc

Where e represents the base of natural logarithms. This equation describes how a quantity (temperature, water, gas, electricity) changes from one value to another (from a to b). On careful examination of the differential equation terms, one can see that the rate of change is proportional to the remaining distance, e.g. as the goal is approached, the rate declines. As it turns out, this equation applies to an amazing number of situations, of which this is just a short list:

In several of the above listed applications (including the electronic and oceanographic examples), a different form of the equation is appropriate, one in which there is a sinusoidal driving waveform. Here is that form.

Version 2:

(4)  y(t) + r c y'(t) = m sin(ω t)

The new terms are: m = the magnitude of the driving waveform, ω = 2 π f and f = frequency of driving waveform in Hertz.


Red = source waveform.
Green = waveform at capacitor in series RC circuit.
This version of the differential equation produces a continuous function with no initial values:

(5)  

Like the form described above, this deceptively simple equation is also widely used in physics and technology to analyze periodic processes, such as the voltage and phase relationships in an electronic RC circuit (see the example on this page) as well as other fields that deal with periodic waveforms. It has even been used by the author to choose a boat departure time from a bay in Alaska. On that occasion it was important to establish the time of slack water in a channel leading from the bay to the ocean. In this case the "driving waveform" was none other than the ocean's daily tidal cycle.

 

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