Is there any closed form solution available for the following equation

Question

I have to find the solution for following equation $$e^{ax^2+ax}=(1+ax)^{ax+1}$$ where $a\geq 0$. I can solve it for $a=1$ but I do not know how to solve it for general values of $a$. Any help in this regard will be much appreciated. Thank in advance.

Answer 1

$x=0$ is a solution for any $a$. This is not the only solution in $\mathbb C$, but I suspect it may be the only solution in $\mathbb R$ if $a > 0$.

Answer 2

By letting $ax=z$ and switching to logarithms the problem boils down to finding a positive solution of

$$ \frac{1}{a}=\frac{(1+z)\log(1+z)-z}{z^2} $$ which exists for any $a>2$ and can be found by Newton's method: the RHS is a decreasing function on $\mathbb{R}^+$ which approximately behaves like $\frac{\frac{1}{2}+\frac{7 z}{30}}{1+\frac{4 z}{5}+\frac{z^2}{10}}$ (this is a Padé approximant at the origin).