I can help you a little in finding the analytic solution. Temporarily consider that y is the independent variable and t is the dependent variable, which would give you the equation:
Integrating with respect to y while making use of partial fractions would lead to:
log(abs(y/(1-y))) = t + C
where C is an arbitrary constant of integration. You would need some initial condition to find its value. The final step would then be to solve this equation for y in terms of t.