Hi:

Is there an algebraic means by which to solve this equation and others like it? I can estimate via Newton's method but I would like to know how to determine solution(s) exactly.

e^x + 2x = 7

Thank you.

Rich B.

Printable View

- Apr 6th 2007, 04:56 AMRich B.solving 'exponential' equation
Hi:

Is there an algebraic means by which to solve this equation and others like it? I can estimate via Newton's method but I would like to know how to determine solution(s) exactly.

e^x + 2x = 7

Thank you.

Rich B. - Apr 6th 2007, 05:21 AMCaptainBlack
- Apr 6th 2007, 10:21 AMThePerfectHacker
I developed a way to solve,

ax+b=e^x if a!=0.

Here.

When my engineering professor boldy announced that there is no way to solve this in closed form. I wanted to show to him that he was wrong. (I love doing that). - Apr 6th 2007, 11:05 PMRich B.
Hi Ron:

I followed you right up to the last line. I don't know what W(...) means. Is that 'function W of the argument 0.5e^3.5'? And if so, what is the function W? Confused...

I will be away for a week, effective Saturday, A.M., so if I don't get right back with the proper 'thanks for filling in the gaps', you will understand I trust.

P.Hckr: Thank you for your response. I gave your work a quick read upon getting in at 1:30 A.M., and I will definitely need a good night's sleep before re-reading and digesting.

Thank you both,

Rich B. - Apr 6th 2007, 11:16 PMCaptainBlack
W is Lambert's W function. W(z) is the inverse function of:

f(w)=w exp(w)

So if z=w exp(w), then w=W(z). It is considered by a number of people that

it ought to be included in the list of elementary transcendental functions.

You can learn more about it here.

That W cannot be written as a finite combination of the other elementary

functions and algebraic operations (as far as I know anyway) shows that

a general equation of the form:

exp(x) + ax + b = 0

cannot be solved using just the normal elemantary function and algebraic

operations in a finite closed form.

RonL - Apr 7th 2007, 06:14 PMThePerfectHacker
- Apr 7th 2007, 06:24 PMJhevon