Need help solving the following differential equation:
$\displaystyle \frac{dN}{dt}=k_1N-k_2N^2-r$, where $\displaystyle N(t_0)=N_0$
It's the $\displaystyle N^2$-term that makes it a bit too difficult for me.
Thanks beforehand!
What have you tried and why do believe it unfruitful?
I haven't really looked at it much, but it seems like a derivative might eliminate the single obstacle to which you refer.
I get $\displaystyle \frac{d^{2}N}{dt^{2}}\;=\;\frac{dN}{dt}(k_{1}-2k_{2}N)$. There, no $\displaystyle N^{2}$ term.
I'm not sure if that's any easier. It's just the first thing that came to mind, based on your suggesting of the sepcific impediment. It seems you'll need another initial value.
As long as it took me to type this, the real and more appropriate solution came to mind. This illustrates an important point. Even if you are wandering off somewhere that leads nowhere, your mind may be stimulated to a better direction.
It's separable.
$\displaystyle \frac{dN}{k_{2}N^{2}\;-\;k_{1}N\;+\;r}\;=\;-dt$
I see completing the square and a trig substitution in your future - and a little alegbra. Show us what you get.