Sorry if this is too easy, but I have really been struggling with this. I have been asked to solve the initial-value problem by separation of variables and determine the maximum time interval of existence of the solutions for $$\dot{x}=x(2-x)^2,\, x(0)=1$$ When integrating, I got $$t=\frac{1}{4}\,\log\left( \frac{x(t)}{2-x(t)}\right)-\frac{1}{2(2-x(t))}+\frac{1}{2}$$ But I need to get an explicit equation for $x(t)$ and I cannot find a way to get it. Should I make a change of variables? Also, I would say that the maximum interval of existence of the solution is $(-\infty,\infty)$?
2026-03-27 01:43:23.1774575803
Initial-value problem by separation of variables
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