1. ## differencial equations

hi

2. Originally Posted by benjaminatkins

A model for the spread of a disease through a population is given by the system:
dx/dt = -4xy
dy/dt = 4xy-500y
Here x(t) represents the susceptible population, and y(t) represents the infected population. Solve the differential equations to find y in terms of x. Sketch and Justify a phase plane diagram.
Discuss what this analysis tells you about the spread of the disease.
Both of these differential equations are separable.
$\displaystyle \frac{dx}{dt} = -4xy$ may be rearranged as
$\displaystyle \frac{dx}{x} = -4y dt$,

and $\displaystyle \frac{dy}{dt} = 4xy-500y$ may be rearranged as
$\displaystyle \frac{dy}{y} = (4x - 500) dt$.

You may also compute $\displaystyle \frac{dy}{dx} = \frac{4xy - 500y}{-4xy}= -1 + \frac{125}{x}$ and then integrate that.