NAME______________________________________ UTEID__________ 427K Uhlenbeck Homework 3 Due 2/10/05 Note that you should be sure that you know how to do this homework for the exam which is scheduled for 2/8/05. Give your answer on this sheet and include your worksheets stapled to it. Leave your answers in terms of logs and exponential, rather than giving numerical answers. 1. Consider the equation y' = y( y^2 - 5 y + 6).a) Is the equation autonomous? b) Find the equilibrium points and classify them as to stability. c) What happens to the solution which starts at y(0)=1? 2. A population of fish in a pond is discovered to have an unstable equilibrium at N = 5,000, a stable equilibrium at N = 10,000. By experiment it is discovered that when the population gets very small, it dies off quickly at the rate of 100 fish per day due to pollution. Find a a model N' = f(N) which can be used as a model for this behavior. Assume f(N) is a quadratic polynomial. 3. Let q' = (q - 2q^2)/(1 + q^2). Determine the stability of the critical point at q = 1/2. Give the linear differential equation which best approximates this equation equation at q = 1/2. If q(0) = .502 what does this approximate equation give for q(2)? 4. I just poured myself a cup of tea 5 minutes ago. The temperature of the water was almost boiling at 200 degrees F. The tea is now drinkable at about 120 degrees F and the temperature of my office is 70 degrees. At what time will the tea become cold too cold to drink (at 85 degrees F). Use Newton's law of cooling, which says that the rate of change of temperature is proportional to the difference between the temperature of the object and the surrounding temperature. 5. A friend of mine just bought a house to rent out for extra income. She is taking out a loan on her own house of $200,000 to buy the house outright. Interest rates on this loan were 8%. Suppose that she wants to pay off the loan in 30 years, and make $200 a month. How much rent should she charge each month?