I found something strange when I try to solve this equatiin of $x$:
$\int_0^t \frac{1}{xW(\frac{1}{xf(\tau)})}d\tau=c_0t$,
where $t$ and $c_0$ are constants. $f(\tau)$ is a known polynomial function. $W(z)$ is the Lambert W function, i.e. $z=W(z)e^{W(z)}$.
If I take the derivative of the equation on both sides, I can get some solution of $x$. However, the result seems wrong since it require for any $x\in (0,t)$, $\frac{1}{xW(\frac{1}{xf(\tau)})}=c_0$ holds. And this is not what I want.
In brief, I have the following two questions:
1, What is wrong with taking the derivative of the equation?
2, Is there any other way to solve $x$?
Thanks!