I have the following differential equation:$$\frac{dx}{dt} = k(a-x)(b-x), 0 < a < b$$.
This is easy enough to solve using a partial fraction decomposition and using logs: I get to $$\left[-\frac{1}{b-a} \ln|a-x| + \frac{1}{a-b}\ln |b-x|\right]_0^x =kt,$$ which can be expressed as $$\frac{1}{a-b} \ln\left(|(a-x)(b-x)|\right) = kt$$ Manipulating this further gives $$|(a-x)(b-x)| = e^{(a-b)kt}$$ How would I isolate $x$ now to get an explicit function $x = x(t)?$
