How can the exponential function y=Aoe^-kt (Ao being the difference between temperature of the substance and its surroundings) be obtained from Newton's Law of Cooling ( dy/dt=-ky)?

Question

kainui · Answer

Simply turn it into an integral by dividing dt from the left and y from the right to get the integrals:

$$\int\limits_{y _{o}}^{y}dy/y=-k \int\limits_{0}^{t}dt$$

kainui · Answer

This yields, ln(y)-ln(yo)=-kt which simplifies by logarithm rules to ln(y/yo)=-kt, put both sides up to a power of e to get y/yo=e^(-kt) and then multiply both sides by yo.

kainui · Answer

Of course, Yo is the same thing as Ao, and here you can see how Ao is just the starting population. Not bad huh?

anonymous · Answer

Awesome, I will give that a try. Thanks for your help!

kainui · Answer

Yeah no problem, if there's anything else I can help with this I'd be happy to.