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Calculus question simplification

I have been working with calculus problems recently and have come across something that, although intuitive, is really hard to prove.
So, this is the situation:
$$\frac{d}{dt}\left(\frac{d\zeta}{dt}\right)=\zeta\frac{d^2\zeta}{dt^2}$$
So, applying the chain rule and integrating I get:
$$\frac{d\zeta}{dt}-\zeta\frac{d^2\zeta}{dt^2}=C$$
I do not really see any way to rearrange it in a simpler form. I have also tried to rearrange the expression by multiplying all the frist terms on the left and all the last terms on the right, but it also seems to be difficult to solve in a similar form.
Does someone have a good idea to prove that this can be simplified?
Thanks!

A:

Some notes on notation:

It’s not $\zeta$ that’s going to be differentiated, its $\frac{d\zeta}{dt}$
Even when this is the case, $\frac{d\zeta}{dt}$ doesn’t usually mean what it is in this equation
$\frac{d\zeta}{dt}$ is usually the derivative of a function $\zeta$ with respect to time, so the most common solution in the literature would be:

$$\frac{d\zeta}{dt}=\zeta \frac{d\zeta}{d\zeta}$$
The left side is a differential, and so far the right side is the derivative of a function of $\zeta$, $\zeta \frac{d\zeta}{d\zeta}$
$\frac{d\zeta}{dt}$ is a rather special case that you
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