How to integrate this differential equation? Applied Maths

Question

dN/dt = aN + bN²

(Where N is variable, a and b are constants)

N(0) = N₀

This is what Im supposed to get:

N(t) = (aN₀) / ((a-bN₀)e^-at+bN₀)

I tried completing the square and a lot of other ways and I just can't do it. Any help would be much appreciated!

Edit: Comes from Logistic Model of Population Growth Equation

Answer 1

This equation is a Bernoulli type ODE where n=2 ... make the substitution v=1/N, thus dN/dt=-1/v^2dv/dt. This changes the ODE to be -1/v^2dv/dt-a/v=b/v^2 that simplifies to dv/dt+av=-b which is a linear first order ODE that can be solved by the Integrating factor method. The result of this is that v=-b/a+Ce^-at ... Now reverse the substitution (v=1/N) yielding N= 1/(-b/a+Ce^-at) where C is a constant that can be found to be 1/N(0)-b/a. This results in N(t) = 1/(-b/a+(1/N(0)-b/a)e^-at)

Answer 2

Hello! After you divide both sides by aN + bN^2 and factor the left-hand denominator into N(a + bN), have you looked into using partial fractions to simplify the integrand? That is how you solve the logistic differential equation! While it would take quite a while to type the solution here, I encourage you to check out some of the many great online resources on logistic differential equations such as:

https://www.khanacademy.org/math/ap-calculus-bc/bc-differential-equations-new/bc-7-9/e/logistic-differential-equation

http://www.cengage.com/resource_uploads/downloads/1133109489_499145.pdf