2. Two differential equations modeling problems follow. Do at least one of them (a) i. If N is the (fixed, constant) population in among residents can often be modeled as follows: Let x = x(t) be the number of people who have heard caught the disease by time t days. Then the disease spreads via the interactions between those who have the disease, and those who don’t. The rate of transmission of the disease is thus proportional to the product of those who have the disease, and those who don’t. Set up a differential equation to model this situation. Use k for your proportionality sufficiently large town, then disease spreading a constant ii. Solve the differential equation you get to find an expression for the number of people who have caught the disease by time t days i. Continuing with this model, when will half the original population have been infected by the disease?