I am working on a growth model for bacteria as a function of a nutrient, and I am stuck. So the differential equation I am supposed to be solving is
$\frac{dN}{dT} = k(C_0 -\alpha N(T)) N$
The problem is here, is that I am supposed to be using partial fractions, and I am unfamiliar with them, except for about one lecture a long time ago. Can someone help me at least get started on where I need to go for partial fractions?
We rewrite your equation as
$$\color{blue}{k} = \frac{1}{N(C_0 - \alpha N)} \frac{dN}{dt} = \color{blue}{\frac{1}{C_0}\left( \frac{1}{N} + \frac{\alpha}{C_0 - \alpha N} \right)\frac{dN}{dt}}$$
Now integrate both sides,
$$\int \color{blue}{k} \ dt = \int \color{blue}{\frac{1}{C_0}\left( \frac{1}{N} + \frac{\alpha}{C_0 - \alpha N} \right)\frac{dN}{dt}} dt = \frac{1}{C_0} \int \left( \frac{1}{N} + \frac{\alpha}{C_0 - \alpha N} \right) dN$$
Can you take it from here?