Question

Find roots of x
e^x+x=4

Answer 1

$e^x +x = 4$

Now we write this in a Lamber form i.e.

$1 = (4-x) . e^{-x}$

Rewrite this introducing u i.e.

let $u = 4 - x$ such that;

$x = -u +4$

Replacing this back into the Lambert form, we have;

$1 = (4 - (-u+4)) . e^{-(-u+4)}$

$1= (\cancel{4} + u - \cancel{4}) .e^{u-4}$

$1 = u.e^{u-4}$ Rewrite this again in lambert form;

$u.e^u = e^4$ Solving this, we have;

$u.e^u = e^4 ⇒ u = W_n (e^4)$

Where W is the Lambert function called the Omega constant

Now substituting this back to $u = - x +4$ and solving for $x$ ;

$⇒ -x +4 = W_n (e^4)$

$\therefore x = - W_n (e^4) +4,$ $n\isin \Z$