Good ODE Books That Explain How Solution Methods Came To Be and Their Justifications As part of the mathematics program offered at my college, I took an introductory ODE course a few semesters back.  This was the one math course in my entire college career that I was totally lost in.  Even after pursuing additional information through my instructor and scouring countless other ODE books, I have no idea how the various methods of solutions for ODEs came to be.  They are all non-obvious to me--well, all except for separation of variables--and really seem quite contrived.
What is a good book or online resource that conveys ODE solution methods in an easy to understand way?  The book I used for my course was "Fundamentals of Ordinary Differential Equations" by Nagle, Saff, and Snider and it was very much written for those people who really only wanted or needed an algorithmic way to find solutions.  What I am really looking for are some explanations for why certain methods work, how they work, and how they came to be from a theoretical standpoint, because right now they all seem like lucky guesses and don't seem interconnected at all.
If you can suggest some books or online resources, that would be great.  Please provide some rationale for why you think the resource is good.  No modeling needs to be covered in the resource as that is rather simple.
 A: I have been in your boat. Trust me you can't go wrong with book
Ordinary Differential Equations, Tenenbaum and Polland. Below is Amazon Link.
http://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407/ref=sr_1_2?ie=UTF8&qid=1389339215&sr=8-2&keywords=ode
It is a fair balance of theory and applications. It is ideal for self study and
as a Dover edition, it is cheap. 
If you feel that is not to your level of rigor, consider
Ordinary Differential Equations, Gian Rota
A: Mostly I prefer , 


*

*Differential equation with application and historical notes by G.F Simmons ( a good book for beginners also it contains a sufficient amount of exercise) 

*Differential equation by S.L Ross 

*An introduction to ordinary differential equations by E.A Coddington ( this is for doing Differential equation rigorously ) 

A: Tanenbaum and Pollard is a very good book. I never used it in a course but I bought it for my library. At 800 pages it covers a lot more than you would normally cover in a one semester Diff Eq course.
