TITLE: The product of the derivative and anti derivative is the function?
QUESTION [8 upvotes]: When is $$f(x) = f'(x)\int{f(x)}dx$$? I just though of the problem but couldn't solve it myself.

REPLY [3 votes]: We can write your equation as follows substituting $y+C_1=\int f(x)$
$$y'=(y+C_1)y''$$
$$y''=y'/(y+C_1)$$
Now integrating each side gives
$$y' = \ln({y+C_1})+C_2$$
$$\frac{1}{\ln{(y+C_1)}+C_2}y'=1$$
And this is as far as we will go.
$$\int \frac{1}{\ln{(y+C_1)}+C_2}dy=x$$
So, the crux of this matter comes to finding $\int \frac{1}{\ln{y}}dy$ for which there is no closed form. From here we would take the inverse of this non closed form function, and then take the derivative of that...
So it didn't work out as nicely as one would hope, but nice thinking. I'd have to agree that this is a cool function, one which is equal  to the product of it's derivative and integral, despite the fact that it is not going to have a nice closed form.