On the web (ie, Wikipedia, and other sites) it seems that exponential decay is always defined as the situation f\;'(t) = -kf(t). However, is not Newton’s Law of Cooling an example of exponential decay? But Newton’s Law of Cooling does not fit this form. What is missing is a constant. If we define exponential decay to be the situation f'(t) = w - kf(t), then everything’s fine. So, is there in fact a constant missing (for convenience?) from the widely-disseminated definition of exponential decay, or, as I suspect, is the definition not all that fixed, depending on some judicious arm-waving as the occasion demands?
What is exponential decay, exactly?
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calculus
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0@Michael Hardy: Bingo. THAT answers my question. I'm shamefaced I didn't notice it myself. Anyway, if you will post your comment as an answer, I will accept it. – 2011-10-19
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With a positive constant $k,$ and other constants $A,B,$ exponential decay is $ f(t) = A + B e^{- k t},$ where you can work out the O.D.E. as you like. Examples include the battery charging curve, also called capacitor, in which $A$ is positive and $B= -A.$
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0I agree, but this is not what Wikipedia and the others say, and it is accounting for the discrepancy that is the point of my question. – 2011-10-19
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Solutions to f'=-kf decay (exponentially) to zero. Your cup of coffee doesn't cool down to zero degrees, it cools down (exponentially) to room temperature, and that's the $w$ of f'=w-kf. If you rejigged your temperature scale to measure degrees above room temperature instead of degrees above zero, Newton Cooling would be f'=-kf.
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0BTW, learning to live with small, well-regulated amounts of ambiguity is part and parcel of what learning a human language (English, French, German, …) is all about. – 2011-10-21