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Unfortunately I have reached the maximum number of math classes I can take for my undergraduate degree. I still wish to study basic ODEs and basic number theory. What is a good textbook with an introduction to these? I would prefer a textbook that is not super rigorous or formal since I will be studying it on my own time.
Thank you.

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Similar questions: books on ODE/PDE math.stackexchange.com/q/3335/823 –  Baudrillard Apr 21 '11 at 8:37
    
Books on introductory number theory: math.stackexchange.com/q/1774/823 –  Baudrillard Apr 21 '11 at 8:38
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Wiki-hammered. For future reference: I highly encourage users of this site to be proactive in flagging this type of questions to the moderator for conversion to community wiki. –  Willie Wong Apr 21 '11 at 11:53
    
@Willie: I have a question for you, I wanted to know what is a community wiki and what is the difference between that and a normal thread? Thanks –  night owl Apr 21 '11 at 12:10
    
@nightowl: please see meta.math.stackexchange.com/questions/445/… –  Willie Wong Apr 21 '11 at 21:42
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6 Answers

I think these two are quite good:

Elementary Differential Equations with Boundary Value Problems (Edwards &Penney)

An Introduction to the Theory of Numbers (G Hardy)

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For Number Theory, Hardy and Wright is a great book, but not a great textbook (no exercises). The book by Joseph Silverman (Friendly Intro. to Number Theory) is fairly small, very nice. Niven, Zuckerman, and Montgomery is a classic text. –  André Nicolas Apr 21 '11 at 5:45
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For learning ODE's, a popular undergraduate book is

1.)Differential Equations with Boundary Value problems-Polking, Bogges, Arnold.

If you want to see many examples, I recommend you get the

2.)Schaum's Outline on Differential equations-Bronson,Costa.

From personal experience I highly recommend both of these books. If you want to see slightly more advance topics with a geometric taste I recommend

3.) Ordinary Differential Equations-V.I. Arnold

As for Number theory, if you want a computational approach, consider

1.) Elementary Number Theory-Burton

For a more theoretical approach,

2.) A classical Introduction to Modern Number theory-Rosen, Ireland 3.) Introduction to Analytic Number Theory-Apostol

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Nonlinear Dynamics and Chaos by Steven Strogatz is a great book if you want to get a feel for how differential equations work. Wonderful explanations, fun exercises, and lots of interesting applications. (But don't expect any proofs of existence and uniqueness theorems and such things.)

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There are two books I know of that deal with differential equations & include a chapter on the calculus of variations at an introductory level as well which you might enjoy.

1) Differential Equations and Their Applications - Zafar Ahsan

2) Differential Equations & the Calculus of Variations - Lev Elsgolts

Another book exclusively devoted to ODE's is Tenenbaum/Pollard Ordinary Differential Equations.

The NPTEL video lectures here are wonderful, do module 1 & 2 simultaneously (beginning with, & with more emphasis on, module 2).

The UCCS video's here use different books, you might like to buy one of the books & work along with the videos on there. Similarly the videos on number theory in that link use a book you might like to buy & read along with the lectures on there.

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@sponsoredwalk: Thanks for the links: I checked out the UCCS site and it seems to want a school email to access the online videos. Do you know a way around this or is this some new implementation that they have created you did not know about? Thanks –  night owl Apr 22 '11 at 3:00
    
No it's fine you just need to give your own e-mail, i.e. just register when they ask you to as you try to access the videos, & you get access. Sorry, should have mentioned that! –  sponsoredwalk Apr 25 '11 at 10:50
    
@sponsoredwalk: Okay thanks for the info, I will do that. –  night owl Apr 26 '11 at 8:09
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Some books that I think work quite well and very well laid out in formatting and examples and exercise are the following two books:

1) Differential Equations & Linear Algebra, Third Edition: Edwards and Penney

2) Elementary Number Theory, Fifth Edition: Kenneth Rosen

There is a sixth edition out now on the Rosen's Number Theory book, but I would guess that there is not much change to it, but I can be unsure. The material should be still relevant in the Fifth with respect to the newer edition.

Okay, I hope that this helps out with your journey to self-learn.

Good~Luck and happy studying. :)

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To save someone else from having to click on the link, the ODE text refer to here is the Edwards & Penney text. –  cch Apr 21 '11 at 15:20
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  1. An Introduction to Ordinary Differential Equations-Earl A. Coddington
  2. Fundamentals of Number theory-William J. Leveque
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