I am practicing for my exam on Friday and I came across a differential equation problem I just can't solve. It's about a disease spreading through a population of $N$ people. At the start ten percent is infected, after one time unit twenty percent are infected. The infection rate is proportional to the amount of people uninfected times the people who are already infected.
These conditions give the following ordinary differential equation for the number of infections $y(t)$ where $t$ is time (let us assume that $t$ has units of days) :
$$
\frac{dy}{dt} = k\,y\,(N – y)
$$
with initial condition
$$
y(0) = \frac{N}{10}
$$
We will also need to use the intermediate condition
$$
y(1) = \frac{N}{5} \,.
$$
I don't know where to go from here. Am I correct when I say it's non-linear?
Best Answer
Yes, it's nonlinear, but separable. Separate and integrate:
$$\frac{dy}{y(N-y)}=kdt$$ $$\int \frac{dy}{y(N-y)}= k \int dt $$ $$\ln(y)-\ln(y-N) = Nkt + C. $$ Use the two conditions given, namely $y(0)=N/10$ and $y(1)=N/5$ to determine $k$ and $C$.
Hope this helps some,
Cheers,
Paul Safier