# Thread: Analyzing solutions of y'= r-ky using W lambert function?

1. ## Analyzing solutions of y'= r-ky using W lambert function?

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 )

2. ## Re: Analyzing solutions of y'= r-ky using W lambert function?

$\displaystyle y = y_0 e^{-kt} + \dfrac{r}{k}(1-e^{-kt}) = e^{-kt}\left(y_0-\dfrac{r}{k}\right)+\dfrac{r}{k}$

$\displaystyle \dfrac{ky-r}{ky_0-r} = e^{-kt}$

$\displaystyle -kt = \ln\left(\dfrac{ky-r}{ky_0-r}\right)$

$\displaystyle t = -\dfrac{1}{k}\ln\left( \dfrac{ky-r}{ky_0-r} \right)$

If $\displaystyle y = 0.99\dfrac{r}{k}$, then plug that into the equation to get:

$\displaystyle t = -\dfrac{1}{k}\ln\left( \dfrac{0.01 r}{r-ky_0} \right)$

Note: you need $\displaystyle \left(r>ky_0\text{ AND }r>0\right)$ or $\displaystyle \left(r<ky_0\text{ AND }r<0\right)$ in order for this to have a real-valued solution.

3. ## Re: Analyzing solutions of y'= r-ky using W lambert function?

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

4. ## Re: Analyzing solutions of y'= r-ky using W lambert function?

Originally Posted by kochibacha
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
One could analyze it without the W Lambert function. There are many methods to analyze a function. The W Lambert function provides a straightforward method to analyze it, but that is by no means the only method.

5. ## Re: Analyzing solutions of y'= r-ky using W lambert function?

Originally Posted by SlipEternal
One could analyze it without the W Lambert function. There are many methods to analyze a function. The W Lambert function provides a straightforward method to analyze it, but that is by no means the only method.
could you give me an example of analyzing variable "k" without using w lambert function from the above problem

6. ## Re: Analyzing solutions of y'= r-ky using W lambert function?

Originally Posted by kochibacha
could you give me an example of analyzing variable "k" without using w lambert function from the above problem
Newton approximation is one of many examples.

7. ## Re: Analyzing solutions of y'= r-ky using W lambert function?

Originally Posted by SlipEternal
Newton approximation is one of many examples.
did you mean newton raphson method?

I think i cannot find k' alone using implicit differentiation
and what variable should I differentiate to respect with

could you recommend some great books where I can study in detail myself

8. ## Re: Analyzing solutions of y'= r-ky using W lambert function?

Originally Posted by kochibacha
did you mean newton raphson method?

I think i cannot find k' alone using implicit differentiation
and what variable should I differentiate to respect with

could you recommend some great books where I can study in detail myself
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.