Solve $\exp(x)(5-x)=5$ by hand

Question

Is there a way to solve this equation by hand?

$\exp(x)(5-x)=5$

Solutions:

$x_1=0$

$x_2= 4.96511$

Answer 1

There is no elementary solution.

You can use the Lambert W-function. This is the inverse of $f(z)=ze^z$.

Substitute in $y=x-5$ to get $e^{y+5}\times -y = 5$. This becomes $e^{y}y = -5e^{-5}$.

We have $y=W_{-1}(-5e^{-5})=-5$ or $y=W_{0}(-5e^{-5})\approx-0.03488$.

Hence $x=0$ or $x\approx4.96511$.

You will need a special calculator to compute the second one.

Alternatively, you can approximate the second solution by a numerical solution, like Newton-Raphson.