Hello,
I am trying to find the analytical solution to this differential equation

dR/dt = k1*(R^k2)*(1-(R/Rmax)); R(0) = Ro

k1 and k2  are parameters that need to fitted, while Ro and Rmax are the
baseline and max value (which can be fitted or fixed). The response (R)
increases
initially at an exponential rate governed by the rate constants k1 and k2.
Response has a S-shaped curve as a function of time and it approaches the
value of Rmax at time approaches infinity.

If there is an analytial solution to this differential equation then it
makes my life easier when trying to perform some non-linear regression.
Kindly provide the integration process so I can learn how to do it myself
for future reference. I believe that the way would be
to use integration by parts (I tried hard to find the solution but keep
getting stuck).

Please help,
Mahesh

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