Math 4791/5791 - Problem Set 3
Fall 1998

Due September 17

1. Suppose the cost of living in a particular country has been increasing at a rate of 3.4% per year since 1990 (t=0). Let denote the 1990 cost of living.
   1. Find the function that gives the cost of living for all t>0. Be sure that your function gives a 3.4% increase each year!
   2. What is the doubling time for the cost of living?
   3. What is the tripling time for the cost of living?
   4. Does the doubling or tripling time depend on ? Explain.
   5. Express the cost of living function using a base of 10 for the exponential function.
2. (based on problems 2.2.3 and 2.2.6 from the book) Analyze the following equations graphically (plotting the phase portrait and direction field), identify the fixed points and discuss their stability. Then sketch a few solutions with different initial conditions.

   

3. (Problem 2.2.8) The phase portrait of an equation of the form x'(t)=f(x) is given below. Find a function f that is consistent with this phase portrait.

   See text for figure.

4. (Problem 2.2.9) Some solutions of an equation of the form x'(t)=f(x) are given below. Find a function f that is consistent with these solutions.

   See text for figure.

5. (based on problem 2.2.13) The velocity v(t) of a skydiver falling to Earth is governed by the equation where m is the mass of the skydiver, g=32 feet per second is the acceleration due to gravity and k>0 is a drag coefficient.
   1. Determine as much information about the solution as possible using graphical methods.
   2. Find the solution analytically subject to the initial condition v(0)=0.
   3. Find the terminal velocity of the skydiver by taking . How does vary with m and k?
   4. Find the function s(t) that gives the position of the skydiver by solving the ODE s'(t)=-v(t) with the initial condition s(0)=0. Sketch the position function carefully.
6. (Problem 2.4.1 and 2.4.7) Carry out the linearized stability analysis for each equilibrium point of the following equations.

   

7. (Coming attraction just to think about for now) A Pursuit Problem: A dog walks north from a crossroads at 1 mile per hour. The dog's master begins one mile east of the crossroads and walks at all times directly at the dog with a speed of s>1 miles per hour. When and where does the master overtake the dog?