I have:

 

[latex] C=\frac{\lambda}{\mu}e^{-({\mu} - {\lambda})t} [/latex]

 

I just want to make lambda the subject, but things got a little messy. After rearranging, differentiating and solving the quadratic I got an explicit solution.

 

I've attached my solution as a jpeg file (was having a nightmare with LaTex).

 

Would anyone care to have a check for mistakes, it got messy so i wouldn't be surprised if there were.

 

 

 

This isn't the solution that came up on WolframAlpha, but maybe it's equivalent?

 

Take a look, maybe it'll illuminate something: http://www.wolframalpha.com/input/?i=c+%3D+%28lambda%2Fu%29*e^%28-%28u-lambda%29t%29+solve+for+lambda.

 

Note that the [math]W_n[/math] function is this http://mathworld.wolfram.com/LambertW-Function.html.

Sato's suggestion that the solution is in terms of the Lambert W-Function or product log is right. Basically you are looking for a solution to

 

[math]z = w e^{w}[/math]

 

and the principal solution is defined as the product log.

If you are still confused then

 

1. multiply both sides by mu

2. expand exponential so it is clear what you are dealing with

3. multiply both sides by e^(mu.t)

4. you now have an equation of the form LHS=x.e^x

5. now you can read up on Lambert W function aka omega function aka product log

 

https://en.wikipedia.org/wiki/Lambert_W_function

Cheers guys, Not come across Lambert's W-Function before - would not have thought up that solution. I'll check it out and let you know how it goes.

 

 

 

 

 

Update: I get the same equation as wolfram now leading to a sensible sounding answer (about what i'd intuitively expect, which is always nice). Luckily all my parameters are positive so avoided that potential mess. Still need to investigate the function though to understand it properly. Thanks again everyone.

Edited August 25, 201510 yr by Prometheus

The problem has changed slightly.

 

C is now a probability at a particular time, t. Is the above still valid?