A friend asked me what are the solutions to this equation.
I know for sure it has 2 solutions (thanks to Desmos) - (0.159, 0.159) and (3.146, 3.146).
I have no idea even where to start solving this problem.
My knowledge in mathematics is sadly high school tier so please keep it as simple as possible.
Thanks in advance!
We need to use the function $W(x)$ which is defined to be the inverse of $xe^x$ on $(-1/e, \infty)$. This is to say, if $xe^x = y$ where $x>-1$, then $W(y)=x$. Note that for $-e^{-1}<x<0$ we have two choices for what $W(x)$ can be. So, if $x < -1$ and $xe^x=y$, we define $W_{-1}$ such that $W_{-1}(y)=x$. We have $$e^{x-2}=x$$ $$\frac 1 x e^x = e^2$$ Letting $u = -x$ gives us $$-e^2=\frac{1}{u}e^{-u} = \frac{1}{ue^u}$$ $$ue^u = -e^{-2}$$ $$\implies u = W(-e^{-2}), W_{-1}(-e^{-2})$$ $$\implies x = -W(-e^{-2}), -W_{-1}(-e^{-2})$$