I can't solve the equation (for variable $\displaystyle E$): $\displaystyle e^{-\frac{E}{T}}*(T+E)=0.01aT$.
Help!
Thanks!
That cannot be solved in terms of elementary functions. You can do this:
Let u= T+ E. Then E= u- T so $\displaystyle \frac{E}{T}= \frac{u}{T}- 1$ and $\displaystyle e^{-\frac{E}{T}}= e^{\frac{u}{T}e^{-T}$
The equation is now $\displaystyle e^{-T}ue^{u/T}= 0.01aT$ or $\displaystyle ue^{u/T}= 0.01aTe^T$.
Now, let v= u/T so that u= TV. The equation becomes $\displaystyle Tve^v= 0.01aTe^T$ or $\displaystyle ve^v= 0.01ae^T$.
Now you can use "Lambert's W function" which is defined as the inverse to the function $\displaystyle f(x)= xe^x$
Then $\displaystyle W(ve^v)= v= W(0.01ae^T)$.
Then, of course, $\displaystyle u= Tv= TW(0.01ae^T)$ and $\displaystyle E= u- T= TW(0.01ae^T)- T or u= T(W(0.01ae^T)- 1)$.
Let $\displaystyle u=T+E$.
Then $\displaystyle E=u-T$, so $\displaystyle \frac{E}{T}=\frac{u}{T}-1$ and $\displaystyle e^{-\frac{E}{T}}=e^{-\frac{u}{T}}*e^1$
The equation is now $\displaystyle e^{-\frac{u}{T}}e^1*u=0.01aT$.
Now, let $\displaystyle v=\frac{u}{T}$, so that $\displaystyle u=Tv$.
The equation becomes $\displaystyle e^{-v}e*Tv=0.01aT$, or $\displaystyle e^{-v}v=0.01a*e^{-1}$.
Forgive me: where am I wrong?