How to find a numerical solution for a differential equation with constraints?

I want first to apologize for my poor english level ^^ Let me present my issue.

I want resolve numerically (I use Python) for x in [0;1] the following differential equation : $$dy/dx = a(y – y^2) $$
with a being a positive constant.

However, i don’t have any initial condition but a constraint which is $displaystyle int_{0}^{1} y(x) , mathrm{d}x = y_{0}$, with 0 < y0 < 1 a constant. The problem is that I don’t know how to add such a constraint.

What I tried to do is to find find a value for $y(0)$ that respects the constraint. However, I think there might be other solution, with other values for $y(0)$, that might also respect the constraint : there isn’t necessarily the uniqueness of the solution.

Is there any known methods to resolve such equations without initial conditions but with such a constraint ? If you have any links that could be useful, I would take it with pleasure.

DSolveValue[{z''[x] == a (z'[x] - z'[x]^2), z[1] == y0}, z, x]
(*Function[{x}, (a y0 - Log[E^a + E^C[1]] + Log[E^(a x) + E^C[1]])/a]*)