Has $e^x = ax^2$ a general solution for all $x$? I was fiddling around with some math and stumbled upon $\exp(x) = a x^2$, finding myself unable to find a solution. Does it even have a general solution $a$ for all $x$? Some googling brought me to the Lambert W function, which I've never encountered before.

Lambert W solution like this: $$ e^x=ax^2 \\\ e^{x/2} = \sqrt{a} \;x \\\ \frac{1}{\sqrt{a}}=x\;e^{-x/2} \\\ -\frac{1}{2\sqrt{a}}=-\frac{x}{2}\;e^{-x/2} \\\ W\left(-\frac{1}{2\sqrt{a}}\right) = -\frac{x}{2} \\\ -2W\left(-\frac{1}{2\sqrt{a}}\right)=x $$ ... and, as Robert notes, also solutions for the other square-root.