I found this equation in one of some slides that I am reading.
$$\frac{dx}{dt}=-\beta xy$$
with solution:
$$x=\frac{n(n+1)}{n+e^{\beta(n+1)t}}\qquad\qquad y=\frac{n+1}{1+ne^{-\beta(n+1)t}}$$
How are these solutions derived?
x and y are defined in this context(there is no other equation except for this):
The missing equation in the OP is the equation on $y$ below. If uninfected people $x$ in contact at a rate $\beta>0$ with infected people $y$ become infected and stay infected, then $$ \left\lbrace\begin{aligned} &\frac{dx}{dt} = -\beta x y \, ,\\ \\ &\frac{dy}{dt} = \beta x y \, . \end{aligned}\right. $$ Therefore, $$ \frac{dx}{dy} = \frac{dx/dt}{dy/dt} = {-1} \, , $$ i.e. $x = x_0 + y_0 - y = n+1-y$. One has to solve $$ \frac{dy}{dt} = \beta \left(n+1 - y\right) y \, , $$ which is a Bernoulli differential equation, with solution $$ y(t) = \frac{\left(n+1\right) y_0 \, e^{(n+1)\beta t}}{y_0 \left(e^{(n+1)\beta t} - 1\right) + n+1} = \frac{n+1}{1 + n e^{-\beta(n+1) t}} \, . $$ Finally, the desired result is obtained.
Note: This is a particular case of the SIR model of epidemiology (see e.g. the Wikipedia page).