Find a solution for an equation

Question

Is there any way to find the solution for $x$ in this equation:

$$ x^2 = e^{2\mu} \left(e^{2x^2} - e^{x^2} \right) $$

Where $\mu$ has a constant value.

I appreciate in advance.

Answer 1

Hint: Consider $$ e^{-2\mu}x^{2}=e^{2x^{2}}-e^{x^{2}}. $$ Let $c=e^{-2\mu}$ and $w=x^{2}$. Then, $$ cw=e^{2w}-e^{w}. $$ Note that if $\mu\geq 0$, $c\leq 1$.

Now, try to show that for $c\leq 1$ and $0 \leq w < \infty$, the curves $cw$ and $e^{2w}-e^{w}$ intersect at only one point: $w=0$.

If, on the other hand, $c>1$, the curves also intersect at some point $w>0$. You will not be able to find a "nice" expression for this root.