Does this equation have any analytical solution?

Question

I am trying to compare my numerically calculated model with analytical solution, which should be provided by following equation

$f(x) = e^{\frac{[f(x)]^2}{2}+ax+b}$

where $a$ and $b$ are constant.

But I don't think that this equation has any analytical solution at all. All I can think of is using some iterative method for every $x$ I need.

Does this equation have any analytical solution or am I stuck with iteration?

Answer 1

For the equation $f=e^{\tfrac{1}{2}f^2 + ax +b}$ Maple gives the solution in terms of the Lambert W function

$$f = e^{ax+b-\tfrac{1}{2}W(-e^{2ax+2b})}$$

Answer 2

Note that $f(x)>0$ and take the square, put all of $f$ to the left side $$ f^2e^{-f^2}=e^{2ax+2b} $$ The Lambert-W function is the inverse to $z\mapsto ze^z$, thus $$ -f^2=W(-e^{2ax+2b})\implies f(x)=\sqrt{-W(-e^{2ax+2b})} $$ For negative arguments in $(-e^{-1},0)$ there are two branches with real values of the Lambert-W function, one with values in $(-1,0)$ the other with values in $(-\infty,-1)$. Depending on your situation you would have to select the correct one. In both cases you need $ax+b<-1$ to get real solutions.

And while the Lambert-W function is analytical, its values are usually computed via an iteration method that solves the defining equation.