Homework 1, due August 30, 2006.

[goodwine]

Unless specified to the contrary, any computer programs and plots may be created using any software that you choose.

Also unless otherwise indicated, all the assigned problems are from the course text, Elementary Differential Equations and Boundary Value Problems, by Boyce and DiPrima, 8th Edition and each problem is worth 10 points.

1. Section 2.1: 4
2. Section 2.2: 25
3. Section 2.4: 32
4. Section 2.6: 22
5. For each of the following, classify the equation as
   • linear or nonlinear (1 point)
   • homogeneous or inhomogeneous (1 point)
   • variable or constant coefficient (1 point)
   • ordinary or partial (1 point)
   and also identify the dependent variable and list the independent variables.
   1. 
   2. 
   3. 
   4. 
   5. 
   6. 
   7. 
   8. 
   9. 
   10. 
   11. 

[drail]

On problem 5i, e = exp(1), correct? If not, what is e (a new variable?)?

Thanks,
David

[goodwine]

On problem 5i, e = exp(1), correct? If not, what is e (a new variable?)?

e equals approximately 2.71828183, so it is a constant, not a new variable.

[goodwine]

Someone asked me:

Do you know of any good, freely available software for plotting and printing direction fields for differential equations such as those found in problem 1 of homework 1? Is it possible to generate direction fields in MATLAB? The extent of my programming experience is limited to that learned last year in the introduction to engineering courses. Thank you for your help.

Unless stated otherwise, hand plots are always fine for this course as long as they are clear and accurate.

Specifically w.r.t. your question, both matlab and mathematica have the capability to plot vector fields like this.

[goodwine]

Someone asked me:

I am having trouble with problem 2.4:32, I will be unable to attend your office hours tom. because I have class. I was wondering if you could give me a few pointers via email on how to solve this problem. I am having trouble figuring out how to treat the g(t) term. I multiplied the equation through by e^2t and do not know how to evaluate the problem from there.

I think the best hint I can give without giving away the answer is to suggest to break the problem up into two time intervals. The only part missing would be the initial conditions for the second part, but that's something you should be able to compute.