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

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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

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## 1 Answer

The equation is separable: $$\frac{dI}{NI-I^2} = a\,dt.$$ To solve it, integrate both sides.

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thank you. Shall I use partial fractions here.. I am sorry, I am kind of stuck. –  Osman Khalid Jan 11 '13 at 0:39
I solved the above equation using partial fractions. I got the same answer I(t) except that there is no 'N' in the denominator... can any one please confirm. Thanks. –  Osman Khalid Jan 11 '13 at 3:04
@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
Thank you again for reply. So do you mean we can multiply N with constant C. Is it possible in maths? Actually, I am not getting N in denominator... kind of frustrated. Is it possible to provide some hints. Thanks in advance. –  Osman Khalid Jan 11 '13 at 22:55
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