Stephen H. answered • 02/26/20

Tutor of Math, Physics and Engineering ... available online

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)