Learning differential equations

[Mathmanman]

What book can I use to learn methods of solutions for differential equations not covered in this textbook:
http://www.barnesandnoble.com/w/fir...7052&ean=9781111827052&isbn=9781111827052&r=1

 

[micromass]

The book seems fairly comprehensive. Usually, more advanced books won't really be about solving ODE's anymore, but rather about showing uniqueness and existence of solutions and about qualitative analysis of the solutions we cannot find analytically.

Why do you think your book is incomplete? What kind of methods would you like to learn?

I really like the following book: https://www.amazon.com/dp/0070575401/?tag=pfamazon01-20 But I don't think there's much in there about finding specific solutions that isn't in the book you linked.

 

[lurflurf]

That is a short basic book, so of course there is a lot more to learn.Since there is so much to learn you should narrow things down a bit. You should also think of what you want to accomplish. Do you want to solve random equations for no particular reason? Do you want to solve equations with some particular application? Do you want to solve equations of general mathematical interest? Are you more interested in exact or approximate methods? Ordinary or partial differential equations?

Some things you might consider
-boundary layer and asymptotic/perturbative methods
-complex variable methods
-numerical methods (your book has some)
-further families of equations
-more with transform methods (I do know any good books on this)
-special and hypergeometric function methods
-more about linear equations with variable coefficients

Here are a few books I have found useful at times, try to flip through them at a library.
Ordinary Differential Equations by Edward L. Ince
general good old fashioned book
Ordinary Differential Equations in the Complex Domain by Einar Hille
complex methods
Elementary Differential Equations by Earl D. Rainville
similar to your book, but with some different topics
Intermediate Differential Equations by Earl D. Rainville
not as good as the first book, but covering more topics
Theory of Differential Equations by Andrew Russell Forsyth
very old fashioned and several volumes
Handbook of Exact Solutions for Ordinary Differential Equations by Valentin F. Zaitsev and Andrei D. Polyanin
gives a good variety of equations
an example
$$x^3\cos^n(x) \mathrm{y}^{\prime\prime\prime}+a\, x^2 \mathrm{y}^{\prime\prime}+x[a-\cos^n(x)]\mathrm{y}^{\prime}+a[a-3\cos^n(x)]\mathrm{y}=0$$
is easy to solve if you know how
can be solved in principle with basic methods you already know
is unlikely to come up for most people

 

[Astronuc]

The book seems fairly comprehensive. Usually, more advanced books won't really be about solving ODE's anymore, but rather about showing uniqueness and existence of solutions and about qualitative analysis of the solutions we cannot find analytically.

Why do you think your book is incomplete? What kind of methods would you like to learn?

I really like the following book: https://www.amazon.com/dp/0070575401/?tag=pfamazon01-20 But I don't think there's much in there about finding specific solutions that isn't in the book you linked.

I used, and still have, the 1972 edition of George Simmons's, Differential Equations with Applications and Historical Notes, which was one of the books in McGraw-Hill's International Series in Pure and Applied Mathematics. The text was revised in 1991, which is cited by micromass. I concur with micromass's assessment.