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I encountered this differential equation in mathematical biology. How can I solve it?

$$ \frac{\partial}{\partial t}F(s,t) = \frac{a}{2}(1-F(s,t))^2 $$

and $$ F(s,0)=s $$

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up vote 6 down vote accepted

Fix $s$ and consider $g(t)=F(s,t)$. Then $g'=\frac12a(1-g)^2$ hence $\frac{g'}{(1-g)^2}=\frac12a$, which yields $\frac1{1-g(t)}=\frac12at+\frac1{1-g(0)}$. Since $g(0)=s$, $$ F(s,t)=\frac{at+(2-at)s}{at+2-ats}=\frac{2s+at(1-s)}{2+at(1-s)}. $$

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