Book to understand how differential equation solutions come about?

[e-pi]

Guys I read Shaum's outline of differential equations. I understand that for different type of differential equation you need different method to solve the equation.

My problem is i am curious how these methods to solve differential equations came about. I want to get a good understanding of differential equations, as probably in the future i hope to derive my own formula to solve non linear water wave ;9

Any good book to recommend me to read? Nothing too rigourous, I am only a engineer with light mathematical training.

Thanks.

 

[qspeechc]

Most people would recommend:
Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard
but I haven't read it. I have read Differential Equations by Shepley L. Ross (2nd Edition) and I think its a very good book. I don't quite know if it's what your looking for. Sometimes a solution or method works just because it does and there's nothing more you can say. Like undetermined coefficients in an exponential to solve a constant coefficient homogeneous linear system. It just works and that's that, nothing more and nothing less.

 

[mathwonk]

tenenbaum and pollard is the best general introduction to ord. diff eq. the best books for in depth understanding are those by v. arnol'd.