# x = y + e^y ? (1 Viewer)

#### milton

##### Member
Given that x = y + e^y
make y the subject.

#### darkliight

##### I ponder, weak and weary
afaik, you can't (using elementary functions).

#### rama_v

##### Active Member
[see image]

Lambert's W function is the inverse of the function wew where w is a complex number. (btw, I don't understand any of this - just putting it into the computer and giving you the result )

http://en.wikipedia.org/wiki/Lambertw

#### Templar

##### P vs NP
LambertW isn't an elementary function. Such a question, if asked, is beyond the scope of the ext 2 (and perhaps even undergrad uni) course.

#### rama_v

##### Active Member
Templar said:
LambertW isn't an elementary function. Such a question, if asked, is beyond the scope of the ext 2 (and perhaps even undergrad uni) course.
Yeah I know, lol was just posting it out of interest more than anything else.

I

#### icycloud

##### Guest
rama_v said:
Actually, it's just y = x - W(e^x)

Here's how to get the answer:

Define the function W(z) as the inverse of y=we^w (i.e. the LambertW function).
Therefore, W(y) = w, where y = we^w.

Now, x = y + e^y
x - y = e^y
1/(x-y) = e^(-y)
e^x / (x-y) = e^(x-y)

Thus, e^x = (x-y) e^(x-y)
Notice we have this function in the form Y = we^w, where w = x-y and Y = e^x

So, W(e^x) = x-y
And we have the answer, y = x-W(e^x)