Hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to solve a problem (related to pharmacokinetics) that requires finding t for an equation like the following:

[itex]A e^{-a t} - B e ^{-b t} + C e^{-c t} = 0[/itex]

where A, a, B, b, C and c are all real positive numbers (they are constants related to the absorption, uptake, efflux and elimination of a substance in the body) and t is time.

a is always different from b.

When c is equal to either a or b, I know how to solve the equation analytically, but that is an exception, it almost never happens.

In the most general case, it appears that there is no analytic solution for this equation (is it true, BTW?).

In such cases I would normally use an iterative method.

However, I have an Excel file where the constants A,a...C can be varied by the user, and I would like t to be recalculated automatically without having to run Solver each time.

So the question is, can an approximate solution (for t) to the above equation be written as a function of the constants A,a...c?

I tried a few things already, e.g. writing y = f(t), Taylor and then inverting, but I always get zero as the solution.

I also searched this forum and the web for similar problems, but I couldn't find anything close (in most cases it seems iterative methods are used).

I suspect the Newton-Puiseux method could be applicable, but I can only find very technical/theoretical descriptions of it, I'm not sure how I could use it in this case.

Any suggestions/links?

Thanks!

L

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# Sum of exponentials: non-iterative approximate solution?

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