AME 302 Homework 3, due February 9, 2005.

[goodwine]

Unless otherwise indicated, all problems are from the course text, Elementary Differential Equations and Boundary Value Problems, by Boyce and DiPrima, 8th Edition.

1. Section 7.8, numbers 7 and 10.
2. Find the general solution to
   • 
   where
   • 
3. Find the general solution to
   • 
   where
   • 
4. Find the general solution to
   • 
   where
   • 
5. Find the general solution to
   • 
   where
   • 
6. Find the general solution to
   • 
   where
   • 
7. Write a computer program (in either C or FORTRAN) to determine a numerical approximation to the solution to the equation in the previous problem if the initial conditions are
   • 
   Using Euler's method is permissible. You must represent the 5 components of the vector as either an array or vector in your program. Also, from your answer to the previous problem, determine the analytical solution to the problem and compare it with your numerical approximate solution.

[goodwine]

A complication regarding computing generalized eigenvectors that I did not anticipate for this homework has arisen. I do not wish to attempt to teach it via this board, so I'm just going to give a hint on how to avoid it.

Whenever you have more than one solution for an eigenvector or generalized eigenvector, i.e., the dimension of the appropriate null space is greater than 1, always pick your eigenvectors to be of the form (1,0,0,0,0)^T, (0,1,0,0,0)^T, etc. This is the case for problems 4, 5 and 6.

This doesn't address the larger issue, but is a way to avoid potential computational complications on this particular homework.

[acrutchf]

when the book asks to sketch a plot of x1 v x2, does it mean that x1 is the top row of the solution or that x1 is the solution associated with c1? Either way, is a parametric plot the way to do this?

[goodwine]

when the book asks to sketch a plot of x1 v x2, does it mean that x1 is the top row of the solution or that x1 is the solution associated with c1? Either way, is a parametric plot the way to do this?

x1 = row 1 and
x2 = row 2.

To plot solutions, you could use ParametricPlot[] in mathematica or just build up a matrix of solutions in matlab inside a Do or For loop.

[kdormuth]

Hi,
Sorry, I'm still a little confused about #7, with the plotting: I've got a file for time and Xi[0]...Xi[4] and I've loaded all that into MatLab. So, what am I plotting? Xi[0[ vs. t? or Xi[0] vs Xi[1]?
Thanks,
Kristin

[goodwine]

Hi,
Sorry, I'm still a little confused about #7, with the plotting: I've got a file for time and Xi[0]...Xi[4] and I've loaded all that into MatLab. So, what am I plotting? Xi[0[ vs. t? or Xi[0] vs Xi[1]?
Thanks,
Kristin

Submit whatever you think suffices to compare the analytical and numerical approximate solutions.