Homework 4

Due Date: Saturday, September 24

Instructions: Feel free to work together with other students in the class, though you must turn in your own copy of the solutions, and you must acknowledge anyone that you worked with. You can turn in your homework assignment by e-mailing me your solutions.

1. Show that, in the West-Brown-Enquist model of blood circulation, the radius of the aorta should be proportional to \(M^{3/8}\), where \(M\) is the total mass of the animal.
2. Let \(y(x)\) be the solution to the following initial-value problem. \[ y' \,=\, 20y^2(1-y)^2,\qquad y(0) = 0.15. \]
   1. Let \(a_n = y(0.1n)\). Write an approximate difference equation satisfied by the sequence \(\{a_n\}\). (Your answer should be an approximate formula for \(a_n\) in terms of \(a_{n-1}\).)
   2. Use Mathematica to implement your difference equation from part (a). Make a Table of values of \(a_n\) for \(n=0,1,\ldots,10\).
   3. Make a ListPlot of the list you generated with the Table in part (b).
   4. Repeat steps (a), (b), and (c) for the sequence \(b_n = y(0.01n)\) with \(n=0,1,\ldots,100\) and the sequence \(c_n = y(0.001n)\) for \(n=0,1,\ldots,1000\).
   5. Use Mathematica's NDSolve command to obtain an approximate solution to the given initial-value problem. Use Plot to draw a graph of this solution.
   6. What were the estimated values of \(y(1)\) according to your approximate sequences \(\{a_n\}\), \(\{b_n\}\), and \(\{c_n\}\)? How do these compare with the value of \(y(1)\) computed by NDSolve?