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