3

Somebody asked this here:

Prove that an equation has no elementary solution

But so far there is no response. The little math I know I have learnt it myself so I dont have a big picture of things. I was wondering if there is a general rule for showing that a function has no analytical solution. For instance:

$$10=x+\ln(x)$$

How can I find the value of $x$? How can I show this is not possible with simple mathematics? Even with no simple math, there is some tool to do it?

Sophie
  • 343
  • 2
    It really depends on exactly what you mean by "analytical". It is often the case that I can't write an answer in terms of the "usual" elementary functions, but can compute it just as well as I can compute sine. – Milo Brandt Feb 05 '15 at 00:08
  • 2
    Closed form is relative term. For instance $\sqrt 2$ is only a closed form because you allow the square root symbol. The number $\sin(1)$ is only closed form because you consider $\sin$ to be one of your standard functions. In your particular equation, the solution is $W\left(e^{10}\right)$ where $W$ is the Lambert function (this is one that not everyone would consider to be an elementary function, but some people find important enough to add it to that class). See special functions for more. – Git Gud Feb 05 '15 at 00:11
  • The value of the solution can be found numerically, not differently for how one would 'find' $\sin(1), e$ or $\pi$. – Git Gud Feb 05 '15 at 00:14

1 Answers1

1

Yes there are no algebraic solutions to this problem frankly I can't answer the last bit of your question however I can demonstrate an algebraic-ish solution. So basically there is something called Lambert W function and the function is shown by a capital W.

$W(c)=x , c= xe^x$

It is a pretty useful function which allows us to solve equations like $x^x=5$. Even though it is not an algebraic solution it still gives us an insight about what we're dealing with.

In the given example we can apply the following like this: $10-lnx=x$

$lne^{10}-lnx=x$

$ln{e^{10} \over x}=lne^x$

${e^{10} \over x} = e^x$

$e^{10} =xe^x$

$W(e^{10})=x\approx 7.9294$

You can find the values for the given input on Lambert W Function by using Wolfram Alpha http://www.wolframalpha.com/input/?x=0&y=0&i=productlog(e%5E10)