Question

2. The growth rate of a population of bacteria is directly proportional to the population p(t) (measured in millions) at time t (measured in hours).

(a) Model this situation using a differential equation.

(b) Find the general solution to the differential equation.

(c) If the number of bacteria in the culture grew from p(0) = 200 to p(24) = 800 in 24 hours, what was the population after the first 12 hours?

3. Find the particular solution y(x) to the following boundary-value problem, y ′′ + 9y = 0, with y(0) = −1 and y(π/6) = 1.

4. Consider the differential equation y ′ + 3t 2 y = e −t 3 with initial condition y(0) = 1. Find the particular solution of the differential equation.

6. Consider a rod: 0 < x < L with insulated sides. The temperature at the sides is fixed at 10 degrees C. At time t = t0, the rod is given an initial temperature distribution of f(x) degrees C, for 0 < x < L. Let u(x, t) denote the temperature at the point x on the rod at time t. The heat flow is modeled by the heat equation, ut − 2uxx = 0.

(a) Write the initial condition(s) of the problem in terms of u(x, t).

(b) Write the boundary condition(s) of the problem in terms of u(x, t).

(c) What does the solution of this problem represent for the rod?

(d) Suppose you are not interested in solving the problem but in finding the steady-state solution. What differential equation would you solve instead?

Answer 1