Power Law from ODE

schaefera

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.

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?

haruspex

Looks to me that c and k are the same as before, but a and b are inverted.
Seems good to me. Why do you think it's invalid? You want to know the relationship between M and R at each given value of t.
You don't?
a*ln(M/C)=b*ln(R/K)
ln((M/C)^a)=ln((R/K) ^b)
(M/C)^a=(R/K) ^b

schaefera

Thank you so much! Silly me- I was exponentiating both sides without bringing the constants into the log.

I was unsure about solving for t because it seemed like I'm setting the two sides equal to each other for all t while they might not always be equal.

Chestermiller

Just divide one differential equation by the other to eliminate the dt.