Differential Equations

Posted on Feb 5, 2014

Let`s examine the statements that describe the equation. In part (1) of the statement, we see that an unknown function y(t) is added to its derivative y`(t), which is scaled by two multiplier terms r and c. Remember about differential equations that, unlike numerical equations, they describe dynamic processes ” things are changing. Remember also

Differential Equations
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that the derivative term y`(t) describes the rate of change in y(t). Please think about this system for a moment. Let`s say that the variable t represents time (although the equation doesn`t require this interpretation). At time zero, the function y(t) equals a, therefore at that moment the derivative term y`(t) is equal to (b - a) / (r * c). Notice that y`(t), which represents the rate of change in y(t), has its largest value at time zero. Because of how the equation is written, we see that the value of y`(t) (the rate of change) becomes proportionally smaller as y(t) becomes larger. Eventually, for some very large value of t, the rate of change represented by y`(t) becomes arbitrarily small, as y(t) approaches the value of b, but never quite gets there. Put very simply, this equation describes a system in which the rate of change in the value of y(t) depends on the remaining difference between y(t) and b, and as that difference decreases, so does the rate of change. As it happens, this equation is used to describe many natural processes, among which are: Electronic circuits consisting of resistors and capacitors (hence the equation`s terms r and c), where the voltage on a capacitor changes in a way that depends on the current flowing through a resistor, and the value of the resistor`s current depends on the voltage on the capacitor. Heat flow between a source of heat energy and a cooler object being heated by it (like a...

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