My question is to find the function $f(t)$ such that $\frac{df}{dt} = -2f(t)\int_{0}^{t}f(s)\, ds$ with $f(0) = 1$.
My idea is to divide both sides by $-2f$ and differentiate both sides, and then let $g = \frac{df}{dt}$ and consider $g$ as a function of $f$ which would reduce the order of the differential equation, but this doesn't seem to be working. Is this the right way? Is there another way to do this?