Math question

[sivullinen]

Hey flist,

how would you solve 2we^(w^2)-2e^w=0?

(And just to make sure, it is the f'(w) of a function f(w)=e^(w^2)-2e^w, right?)

I hope there's at least one math geek among you :S

ps. I usually love this kind of math problems, but for some reason now I can't do it :(

Comments

[thephoenixboy]

Directed here by aiwritingfic.

Looking through the other comments, my method was much the same as shown on her picture, with a similar result. The only solution I could find algebraic was w=1 - which could very well be the only solution. I then stuck into into my maths package on my laptop. It's possible that I coded it wrong, but either way it didn't come out with anything more helpful so we could well be right.

If you're using that eqn to find max/min, you're not expecting a whole string of answers for an eqn involving e.

Sorry I couldn't be of more help

[sivullinen]

The only solution I could find algebraic was w=1 - which could very well be the only solution.

Yup, it is. So, did you have some actual way for getting it? Or was it just a conclusion reached by looking at it, or something?

Thank you, in any case! ^^

[thephoenixboy]

Ok

Let y=e^w cos it makes it easier to look at

2wy^w-2y=0
Divide by 2
w(y^w)-y=0
Take out a factor of y
y(wy^(w-1)-1)=0
So either y=0 or w(y^(w-1))=1

Replace y by e^w
e^w=0 never happens, so no solutions from this factor

Rearranging the other equation
y^(w-1)=(1/w)
By recognition, if you take w=1, y^(w-1)=1 because anything to the power 0 is equal to 1. Also 1/1=1, so w=1 is a solution.

I've played around with the second factor some more and can tell that there aren't any other integer solutions but off the top of my head I can't prove that there aren't any non-integer solutions.

[macey_muse]

Argh yes, of course ~ I got to w = e^(1-w) (slightly different route, and destination) which of course, w has to equal 1. Anything bigger and RHS gets fractional, smaller and we get weird roots of e and strangeness. (Although, lack of -proof- makes me twitch)

[thephoenixboy]

I'd like to prove it as well. I can see that it's true and I can see why it's true, I just can't prove it.

Still, good to know that we agree :D

[sivullinen]

I love how you can tell someone is a real math lover when they aren't content to just have a solution but really want to know why ;)

Huge thanks for all your help! :)

[thephoenixboy]

I think it's more that people have come up with counterexamples too often for us to trust anything that isn't a proof.

No problem with the help - aiwritingfic said 'maths' and I was over here like a shot :D