I'm trying to find a power law relationship between mass and metabolic rate, given that each of these quantities is defined by a differential equation.(adsbygoogle = window.adsbygoogle || []).push({});

Assuming dM/dt=a*M(t) and dR/dt=b*R(t), where M(t) is mass and R(t) is metabolic rate, I know that I can solve each of these equations to get:

M(t)=c*exp[at] and R(t)=k*exp[bt].

Here is where the first part of my question comes in: Let's say I try to solve both of these for t, and then set them equal to each other. Then I end up with a*ln(M/C)=b*ln(R/K) where C and K are different constants than before, but that's not really important. Is solving for t and setting equal allowed? I'm not sure if I'm looking at a specific time where the two equations are equal in this case, but I can't think of any other way to get rid of the variable.

Otherwise, I would think of dividing the two equations and getting M(t)/R(t)=h*exp[(a-b)t] where h is, again, a new constant that is unimportant. In this case, M=h*R*exp[(a-b)t]... which is different than when I eliminate t.

In either case, I don't see a power law relationship! These are exponential, and not power law, equations unless I'm very mistaken. How can I get to the final product to see the power law in play?

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# Power Law from ODE

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