1

I am trying to solve the equation:

$-\frac{x}{\gamma} = \ln\left( \frac{1}{2}x + \lambda\right)$

for $x$. $\lambda$ and $\gamma$ are constants. By using,

$e^y = x \leftrightarrow \ln x = y$,

I get,

$e^{-\frac{x}{\gamma}} = \frac{1}{2}x + \lambda$

$e^{-\frac{x}{\gamma}}-\frac{1}{2}x= \lambda$,

where I am stuck on how to solve for $x$. Can this equation be solved with a closed form solution for $x$ ? Much help/suggestion is appreciated.

tarmizi
  • 23

1 Answers1

2

$e^{-\frac{x}{\gamma}}-\frac{1}{2}x= \lambda$,

where I am stuck on how to solve for $x$. Can this equation be solved with a closed form solution for $x$ ? Much help/suggestion is appreciated.

It can't, unless you allow special functions such as the Lambert-W-function (see here).

Can you elaborate more on "allow special function" ? for the solution of the equation ?

Yes. You should look up the documentation of that special function (follow the link above) to see what it does. It allows to give an explicit form of the solution, but in a sense this is hiding (or 'moving') the problem in that special function. If this is useful or not, depends on what your goal is.

If we don't allow such functions, that means there is no solution for x ?

No, there's a difference between "no solution" and "no explicit, closed form". Certain equations can have one or more solutions, without the possibility of giving them in a closed form. That doesn't mean the solutions don't exist, you could e.g. find them numerically or graphically.

Community
  • 1
StackTD
  • 27,903
  • 34
  • 63
  • Can you elaborate more on "allow special function" ? for the solution of the equation ? If we don't allow such functions, that means there is no solution for x ? – tarmizi Apr 04 '17 at 07:40
  • @tarmizi I updated my answer. – StackTD Apr 04 '17 at 07:44
  • Thank you very much on the explanation. This is the first time I've heard about the Lambert-W-function. – tarmizi Apr 04 '17 at 07:53
  • @tarmizi You're welcome! – StackTD Apr 04 '17 at 07:54
  • @tarmizi. This is a beautiful function to learn about. Just for your information, any equation which can write $A+Bx+C\log(D+Ex)=0$ has explicit solution(s) in terms of Lambert function in the real and complex domains. – Claude Leibovici Apr 04 '17 at 08:11