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
The Laplace transform is useful because
- it transforms an ODE into an algebraic equation in the transformed variable (or a PDE into an ODE), and
- it includes the initial conditions as part of the algebraic equation.
The usefulness of this LT of course depends on the ability to find the inverse transform of the solution to the algebraic equation. In general, this requires the evaluation of a complex integral. However, for those not versed in complex integration methods, there are tables from which one may simply write down the inverse. Thus, the LT provides a simple means of solving an ODE using algebra.
No better way to understand this than to provide a specific example. Let's solve
$$f''(t) + 4 f'(t) + 3 f(t) = t \sin{t}$$
$$f(0) = 0 \quad f'(0) = 1$$
The LT of the LHS of the equation is
$$s^2 F(s) - 1 + 4 s F(s) + 3F(s) $$
The LT of the RHS is
$$\int_0^{\infty} dt \, t \sin{t}\; e^{-s t} = \operatorname{Im}{\int_0^{\infty} dt \, t \, e^{-(s-i) t}} = \operatorname{Im}{\left [\frac1{(s-i)^2}\right ]} = \frac{2 s}{(s^2+1)^2}$$
We need only solve a simple algebraic equation to find $F(s)$:
$$F(s) = \frac{(s^2+1)^2+2 s}{(s^2+1)^2 (s+3)(s+1)} $$
We may then use tables or the residue theorem to find the inverse:
$$f(t) = \frac14 e^{-t} - \frac{47}{100} e^{-3 t} + \frac{1}{50} [(5 t+2) \sin{t} +(11-10 t) \cos{t}]$$
Best Answer
From the MAA review of Differential Equations with Applications and Historical Notes:
From the MAA review of Differential Equations: Theory, Technique, and Practice:
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.