Eh? How do I solve xe^(a/x) = b?

  • Thread starter Thread starter WarPhalange
  • Start date Start date
AI Thread Summary
The equation xe^(a/x) = b cannot be solved algebraically due to its transcendental nature, requiring numerical methods for a solution. The problem relates to charge carrier density in semiconductors, leading to a similar equation T*exp(-a/T) = b. Despite the complexity, the approximate solution is around 520. The discussion highlights the impracticality of finding an exact solution, emphasizing the necessity of numerical approximation methods. Ultimately, the Lambert W function is mentioned as a theoretical tool, but it still necessitates numerical computation for practical use.
WarPhalange
Eh? How do I solve xe^(a/x) = b?

Homework Statement



xe^(-a/x) = b, where a and b are numbers that are given and I'm trying to solve for x.

The Attempt at a Solution



All I can think of are Taylor series, which won't work in this case because a is ~4000 and b = ~0.1, so I'd need to expand to a lot of terms.

I could try to do it numerically, but would rather not... I know the answer ends up being ~520.
 
Physics news on Phys.org


You have a transcendental equation here, with no algebraic solution. Are you sure this is the correct equation?
 


Yes. I'm trying to solve the equation that's in the book.

It's for charge carrier density in a semi-conductor

n^2 = B*T^3*exp(-E/kT)

Where n, B, E, and k are constants that are given. So I divide by B, then cube root it to obtain T*exp(-a/T) = b

A similar problem is later on in the homework set, where the professor says to calculate it numerically. I mean, the problem is identical except you first have to find n with another equation, then you're back to that. But he makes no mention of calculating it numerically here.

I think, though, that he still wants us to, because like you said, no easy solution. It would take a team of Russian mathematicians in one of Russia's finest Gulag's at least a year to find the solution*. I'll just do this numerically also. In the least I needed confirmation that it's not a trivial answer. Thanks.

*I'm taking a shot in the dark here.
 


There is actually no possible way of expressing that number in terms of commonly known constants. It would only take, however, someone with a big ego typing onto Physicsforums to define, eg, The GibZ Constant, whose definition is the unique solution to the above equation, numerically approximately *blah blah*. Those are your options, but in the end, for a "useful" answer you'll need a numerical approximation, no way around it.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
View full post »

Thread 'Making the denominator of a certain fraction real'

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
View full post »

Similar threads

Solve 2 Triangle Problem: Calculate Exact Value of x
Replies
7
Views
2K
Solving system of 3 quadratic equations
Replies
4
Views
2K
Express ##x^3+y^3## in terms of ##m## and ##n##
Replies
17
Views
3K
Solve for ##x## involving modulus
Replies
10
Views
2K
How Do You Solve the Inequality ##x^2<4## for Negative Values of ##x##?
Replies
6
Views
2K
Solve for ##x## in the Given Problem
Replies
17
Views
1K
Proving a sum of three squared terms, cyclic in #a,b,c#, is equal to 1
Replies
16
Views
2K
Solving an equation for ##x## involving inverse circular functions
Replies
15
Views
2K
What Is the Lambert W Function and How Does It Solve Transcendental Equations?
Replies
6
Views
3K
How Can I Solve a System of Equations With Complex Numbers?
Replies
19
Views
2K

Hot Threads

Recent Insights

Back
Top