Differential Equation (Application)

A population of mice, initially numbering 30, is kept in conditions that can support a population size of 120.  The rate of increase in the population of size N is assumed to be given by the equation \displaystyle \frac{dN}{dt}=cN\left(K-N\right), where c and K are positive constants.  Find the formula giving the number of mice t months later, if, after one month, the population numbers 80.

Source: Rosenberg, R.L. (1984). Elementary Calculus: Course Notes. Ottawa, Canada: Holt, Rinehart and Winston. (Exercise 44: Question 5, p. 132)

Differential Equation (Application)

The growth of a cell depends on the flow of nutrients through its surface.  Let W(t) be the weight of the cell at time t.  Assume that for a limited time the growth rate \displaystyle {dW}{dt} is proportional to W^{\frac{2}{3}}.  (If the density remains constant, then W is proportional to D^{3}, where D is the diameter of the cell, and the surface area is proportional to D^{2}, or equivalently W^{\frac{2}{3}).

Hence \displaystyle \frac{dW}{dt}=kW^{\frac{2}{3}} for some positive constant k.  Find the general solution to this differential equation.

Source: Rosenberg, R.L. (1984). Elementary Calculus: Course Notes. Ottawa, Canada: Holt, Rinehart and Winston. (Exercise 44: Question 4, p. 132)

Differential Equation

The rate at which weight is lost by an animal suffering from a viral infection is observed to satisfy the relationship \displaystyle \frac{dy}{dx} = \frac{t}{72}\left(8-t\right)^{\frac{1}{3}} for 0 \leq t \leq 8.  Here L is the weight loose, as a percentage of the animals original weight, t weeks from the time of infection.  Find the function giving the percentage weight loss L at time t, and determine the percentage weight loss when the animal dies eight weeks later.

Source: Rosenberg, R.L. (1984). Elementary Calculus: Course Notes. Ottawa, Canada: Holt, Rinehart and Winston. (Exercise 44: Question 3, p. 132)