91Ó°ÊÓ

(a) Suppose that a quantity \(y=y(t)\) increases at a rate that is proportional to the square of the amount present, and suppose that at time \(t=0 .\) the amount present is \(y_{0}\) Find an initial-value problem whose solution is \(y(t)\) (b) Suppose that a quantity \(y=y(t)\) decreases at a rate that is proportional to the square of the amount present, and suppose that at a time \(t=0 .\) the amount present is \(y_{0}\) Find an initial-value problem whose solution is \(y(t)\)

A cell of the bacterium \(E .\) coli divides into two cells every 20 minutes when placed in a nutrient culture. Let \(y=y(t)\) be the number of cells that are present \(t\) minutes after a single cell is placed in the culture. Assume that the growth of the bacteria is approximated by a continuous exponential growth model. (a) Find an initial-value problem whose solution is \(y(t)\) (b) Find a formula for \(y(t)\) (c) How many cells are present after 2 hours? (d) How long does it take for the number of cells to reach \(1.000,000 ?\)

Polonium-210 is a radioactive element with a half-life of 140 days. Assume that 10 milligrams of the element are placed in a lead container and that \(y(t)\) is the number of milligrams present \(t\) days later. (a) Find an initial-value problem whose solution is \(y(t)\) (b) Find a formula for \(y(t)\) (c) How many milligrams will be present after 10 weeks? (d) How long will it taked for \(70 \%\) of the original sample to decay?

Solve the differential equation by the method of integrating factors. \(2 \frac{d y}{d x}+4 y=1\)