How can I find solution of this equation? Equation: $$Ge^{-G} = \frac{1-G}{e-1} $$
I like to find solution $G$ of the equation.
Please give some direction 
Thanks
 A: This kind of equation does not show solutions in any closed form and numercical methods need to be used.
You could use it as it is written or, using componendo and dividendo, rewrite it as $$f(G)=e^G (G-1)+(e-1) G=0$$ It first derivative $$f'(G)=e^G G+e-1$$ is always positive and then $f(G)$ is an increasing function.
To find the root, let us use Newton method which, starting at a "reasonable" guess $G_0$ will update it according to $$G_{n+1}=G_n-\frac{f(G_n)}{f'(G_n)}$$ Let us be very lazy and start iterating from $G_0=0$; the iterates will then be $0.581977$, $0.490695$, $0.486239$, $0.486230$ which is the solution for six significant figures (you could notice that we had one overshoot of the solution; this is normal since $f(0) f''(0)=-1 <0$ - Darboux theorem).
Using higher order methods such as Halley (cubic convergence), still starting at $G_0=0$, the  iterates would have been $0.483420$, $0.486230$.
Being less lazy and having looked at the plot of the function, we could have developed $f(G)$ as a Taylor series at $G=\frac 12$ and get $$f(G)=\frac{1}{2} \left(-1-\sqrt{e}+e\right)+\left(-1+\frac{\sqrt{e}}{2}+e\right)
   \left(G-\frac{1}{2}\right)+O\left(\left(G-\frac{1}{2}\right)^2\right)$$ from which the solution would be $$G\approx \frac{3 \sqrt{e}}{2 \left(2e+\sqrt{e} -2\right)}\approx 0.486321$$ This would have been the first iterate of Newton method satrting at $G_0=\frac 12$ and the convergence would have been quite faster (only one extra iteration). Pushing Taylor expansion to second order and solving the quadratic would give as estimate $\approx 0.486229$.
Edit
May be, you could be amazed to hear that using the simplest Pade approximant (instead of Taylor), we should get $$f(G)\approx \frac{a_0+a_1(G-G_0)}{1+b_1(G-G_0)}$$ (I shall not report the expression of the coefficients). Doing it at $G_0=0$  the approximate solution would be $$G\approx \frac{2 e-2}{3-4 e+2 e^2}\approx 0.497693$$ while doing it at $G_0=\frac 12$ the approximate solution would be $$G \approx \frac{9 \left(-\sqrt{e}+e+e^{3/2}\right)}{2 \left(8-5 \sqrt{e}-11 e+5 e^{3/2}+8
   e^2\right)}\approx 0.486230$$ which is the solution for six significant figures.
A: $\frac{G}{e^G}=\frac{1-G}{e-1} \implies \frac{e^G}{e-1}=\frac{G}{1-G}$. 
Now applying componendo and dividendo, we get $\frac{e^G-e+1}{e^G+e-1}=2G-1 \implies G=\frac{e^G}{e^G+e-1}\implies $
$(G-1)e^G+Ge-G=0 \implies G \approx 0.48623$ (I used wolfram here for this equation))

