My instructor used these notes for the lectures of a basic theory of ODE course. Personally, I'm not a huge fan of the notes, but it does cover the uniqueness and existence (e.g. Gronwall's inequality) theorems pretty well and has a good mixture of computational and more theoretical exercises. The notes are based off of Ordinary Differential Equations By Birkhoff and Rota
From the MAA review of Differential Equations with Applications and Historical Notes:
Some years ago, an attempt was made to update Simmon’s book. The result was published as Differential Equations: Theory, Technique, and Practice, by Simmons and Steven Krantz. Alas, much of the charm of the original disappeared in the new version. So it is good news that CRC has brought back the original book in a third edition. I compared it to the second edition and decided that the changes are mostly minor additions dealing with topics Simmons enjoys. Most importantly, the author’s unique personality shines through.
From the MAA review of Differential Equations: Theory, Technique, and Practice:
Differential Equations: Theory, Technique and Practice is an introductory text in differential equations appropriate for students who have studied calculus. It is based on George Simmons' classic text Differential Equations with Applications and Historical Notes. The preface says that this revised version brings the older text up to date and adds some more timely material while streamlining the exposition in places and augmenting other parts.
While this is a more than adequate introductory book on differential equations, it is rather a disappointment—especially given its heritage. I say this knowing that the revised text is, in several places, pedagogically superior to the original book. Nonetheless, Simmons' classic text has a kind of charm that is still apparent 34 years after its original publication. Its pervading sense of reverence for scholarship and respect for work of past masters were uncommon when the book first appeared and are now rare indeed. Some of Simmons' touch remains, but it seems woefully diluted.
For more (opinionated) information on the similarities and differences, I think you will find the full text of the second review to be helpful.
Clarification: The first review is of Differential Equations with Applications and Historical Notes. The book has three editions. It seems that the second edition added a lot but the third edition didn't add that much. The third edition was published in 2018, so it is clear that this book has not been abandoned, despite the release of the second book in 2007. The second review is of Differential Equations: Theory, Technique and Practice, which is a different book. This book is not another edition of the first one; it is a different book based on the first one, sharing one of the authors. The opinion of the reviews in general is that the first book is charming and but conflicts with modern ideas about what should be taught in Differential Equations (see this article for some of the criticisms levied against the traditional style; in general the older style stresses finding analytic solutions and the new style stresses qualitative and numerical methods). The second book is a bit more in line with the newer style, but the reviewers complain that it has lost the charm of the first one. Ultimately, which book is the "best" is an opinion. Personally, I am going to read the first book because it seems more fun, but it is important to keep in mind that in the "real world," some of the more specific methods to find analytic solutions (such as integrating factors) don't see much use.
Best Answer
For a text with a solution manual: See P. Blanchard, R. L. Devaney, G. R. Hall, Differential Equations, 4th Ed., 2011.
Another, less expensive choice ("cheap" compared to most textbooks), is Ordinary Differential Equations, from Dover Books on Mathematics collection. The book has received great reviews, and includes solutions to most of the exercises.
For another "cheap" reference, see Schaum's Outline of Differential Equations, 3Ed. The link will take you to Amazon.com where you can "preview" the book, it's table of contents, etc..
The following are not "text books" in the usual sense of the term, but
Paul's Online Notes may help elucidate some of the concepts you're struggling with, and includes many worked examples, with solutions.
See also Khan Academy: Differential Equations, where you can access tutorials, and accompanying exercises with solutions.