Dear 293: 9/17/96
MATH 293, Homework 4 Due Wed Sept 25, 1996 in lecture.
No hard copy of this assignment is being passed out.
Please read the course homework information and follow the directions there.
You can read the latest version of such by following the links from
http://tam.cornell.edu/faculty/ruina/www_293/home.html
1. A group of people are given the following IVP: y' = y sin(x), y(pi/2) = 3.
Here are the solutions that they find by some means or other.
Which of them are right (could be none or several) and why? (Variation of
parameters and integrating factors are not allowed in your explanation).
a) y = 3 * (cos(x))^2
b) y = exp( - cos(x) )
c) y = 3 * sin(x)
d) y = 3 * exp(-cos(x))
2. Make up a word problem and corresponding ODE (not from book or lecture) that
has an equilibrium solution. Show how the words lead to the equation.
Show that the equation has an equilibrium solution.
By any means you find convincing, show whether the equilibrium solution is stable or
unstable.
*. You should be able to do most of the problems 2.3.1-22 (Some require
relatively hard integrals, however). Don't hand any in.
3. In some sense Euler's method is THE way of understanding what
an initial value problem is all about (even if you are never going to use
numerical methods). You should understand Euler's method so that
you can generate the formulas, not just remember them.
For a simple differential equation, initial condition, and range of time
of your choice.
a) Find an analytic solution.
b) Find an approximate solution using Euler's method (do this twice
with two different step sizes).
[You will need to learn `for' loops by, say, typing '> help for'
or using Pratap's book.]
c) Plot these solutions on a common plot and compare the plots
d) Compare the values (say, to several digits) at the final time.
How does the error in Euler's method depend on step size in
your comparison.