# Solution of first-order differential equation $\frac{\mathrm dI}{\mathrm dt}=aNI(t)-aI^2(t)$ [duplicate]

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How do you solve the Initial value probelm $dp/dt = 10p(1-p), p(0)=0.1$?

I am reading a proceeding paper where I encountered this differential equation. Can any one kindly write steps of solution (given below) of this equation. $$\frac{\mathrm dI}{\mathrm dt}=aNI(t)-aI^2(t)$$ This first order ordinary differential equation has the following general solution: $$I(t)=\frac N{1+CNe^{-aNt}}$$

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## marked as duplicate by Hans Lundmark, Stefan Hansen, QiL, Davide Giraudo, Ittay WeissJan 11 '13 at 9:42

The equation is separable: $$\frac{dI}{NI-I^2} = a\,dt.$$ To solve it, integrate both sides.
@OsmanKhalid There should be an $N$ in the denominator, i.e. the form given in the question is correct. –  mrf Jan 11 '13 at 7:12
@OsmanKhalid (But if you like, you can include it in the constant, of course: rename $C$ to $CN$.) –  mrf Jan 11 '13 at 7:24