I was wondering if a closed form solution does exist for the following equation:
a^2 Exp[x/a] + x^2/2 (1 - b) - a^2 - a x Exp[x/a] = 0
where a and b are Real numbers. I would like to obtain a solution of the form x=f(a,b).
I tried to use the following command:
Solve[a^2*Exp[x/a] + x^2/2*(1 - b) - a^2 - a*x*Exp[x/a] == 0, x,
Assumptions -> Element[a, Reals], Assumptions -> Element[b, Reals]]
but I obtained the following message: Solve: This system cannot be solved with the methods available to Solve.
I was wondering whether I could use the Lambert W function to solve this equation or if there is any way to obtain an approximated solution of this equation. I am looking for a solution for x<0, and I know that a>0, if this can make any difference.
Thank you.
Edited after the 1st reply:
The last sentence is probably superfluous: I am interested in the general closed-form solution of this equation. Then, I will use the part of solution when x<0 (which happens when 0<b<1).