I'm very interested in Lambert $\operatorname{W}$ function and I want to know how to check if some equation can be solved using this function.
Example $1$:
$$e^xx=a$$
For this equation it is obviously that $x=\operatorname{W}_k(a)$ where $k\in\mathbb{Z}$.
Example $2$:
$$a^xx=b$$
Now we must reduce it to form $e^{f(x)}f(x)=c$ and then use Lambert $\operatorname{W}$ function.
$$e^{x\ln a}x=b$$
$$e^{x\ln a}x\ln a=b\ln a$$
$$x\ln a=\operatorname{W}_k(b\ln a)$$
$$x=\dfrac{\operatorname{W}_k(b\ln a)}{\ln a}$$
This is not too hard to solve.
Example $3$:
$$a^x+bx+c=0$$
It is very hard to solve this and after long computation we will get
$$x=\dfrac{-b\operatorname{W}_k\left(\dfrac{\ln a\cdot a^{-\dfrac{c}{b}}}{b}\right)-c\ln a}{b\ln a}$$
Example $4$:
$$a^{x^2}+bx+c=0$$
There is no known solution for this equation.
Example $5$:
$$\sin x+x=a$$
or
$$\dfrac i 2e^{-ix}-\dfrac i 2e^{ix}+x=a$$
My question is how to check if some equation can be solved using Lambert $\operatorname{W}$ function.
