I want to ask a similar question as found here: Solving an ODE with non-local coefficient. Except my ODE is slightly more complicated because the integral is not constant over $x$. Consider this equation:
$$\frac{dy}{dx}(x) = \left(1 - \int_0^xy(x')dx'\right)y(x)$$
If necessary, the solution only has to exist on an interval, say $[0,1]$. Is there any way to attack this analytically? The question I linked only has one answer which is just to guess a solution. What if we change $y$ into $y = \frac{dz}{dx}$ and then break up the definite integral? I think we get,
$$\frac{d^2z}{dx^2} = \left(1 - z(x) + z(0)\right)\frac{dz}{dx}$$
So now we just have to deal with that $z(0)$. If we assume we know $z(0)$, it's possible to get a solution (e.g. if it's some constant, the solution is fairly simple). What if we don't know?