I am trying to solve the function
$$\frac{\partial \left \langle f^{2} \right \rangle}{\partial t} = \frac{\left \langle f \right \rangle - \left \langle f^{2} \right \rangle} {N}$$
where $\left \langle f \right \rangle = p$ and $p$ is a real number between 0 and 1.
$\left \langle f \right \rangle$ and $\left \langle f^{2} \right \rangle$ are the first and second moments of the following equation
$$\frac{\partial f}{\partial t} = \sqrt{\frac{f(1-f)}{N}} \eta (t)$$
where $\eta (t)$ is a Brownian noise term with $\left \langle \eta (t) \right \rangle = 0$ and $\left \langle \eta (t)\eta (t') \right \rangle = \delta(t-t')$ where $\delta$ is a Dirac delta function.
I'm trying to teach myself stochastic processes and performing an integral over the expected value of the second moment is tripping me up.
Advice is appreciated. I did not find a similar question elsewhere on stack-exchang.