$\frac{I}{I-50000}=Ce^{kt}$
Where the constants C and k can be deduced from known values. However I am unable to solve for $I$ in a way which leaves a workable function
2026-02-23 11:39:40.1771846780
Logistic-growth equation rate of change
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Your equation $$\frac{dI}{dt} \propto I(50000-I)$$ is a separable equation.
$$ \frac {dI}{I(50000-I)}=cdt$$
$$ \int \frac {dI}{I(50000-I)}=c\int dt$$
Partial fraction $$\int (\frac {1}{I}+\frac {1}{50000-I})dI=50000ct+k$$
$$\ln \frac {I}{50000-I} = 50000ct+k$$
$$ \frac {I}{50000-I}=Ke^{rt}$$
Cross multiply and solve for $I$ to get $$ I = \frac {50000Ke^{rt}}{1+Ke^{rt}} $$