If , then plug that into the equation to get:
Note: you need or in order for this to have a real-valued solution.
consider this IVP
y'=r-ky , y(0)=y0
y= (y0)e^(-kt) + (r/k)(1-e^(-kt))
if y,y0,r,t are provided, we should be able to solve for k and that's the problem but what I'm really interested is analyzing this problem
if we let y=0.99 (r/k) find t in terms of all other variables
Of courses, if y0 = 0 we can see that t= -(ln 0.01)/k
i wonder if y0 is not zero is it possible to analyze variable t using any knowledge or technology from mathematics?
this problems is derived from real application of pharmacokinetics IV infusion where
y= amount of drug at time t
y0 = initial amount of drug at t = 0
r = infusion rate
k = elimination rate constant
(r/k) = amount of drug at steady-state (as t --> infinity the amount of drug will approach this value and 99% of (r/k) is a good approximation of amount of drug at steady-state)
so what I really ask is how the initial amount of drug reflects the time to reach steady-state ( for example. how much we increase y0 in order to halve the time to reach steady-state )
so it does not require W lambert function in order to analyze this problem?
but apparently one cannot solve for k if other variables are provided with out using w lambert function
I would recommend using the Lambert W function. You were simply asking if there were other methods of analyzing the function, and there are. The easiest method is probably the direct method, which would be the Lambert function. Any book on real analysis should provide some insight on other methods of evaluating functions of multiple variables.