I love R.Courant and F.John's Introduction to Calculus and Analysis because of its wide coverage, precise description and friendly written style.
Are there any classic textbooks like it on ODE/PDE?
Since you like the writing style of Courant and John... for PDEs there are of course F. John's Partial Differential Equations, and also Courant and Hilbert's Methods of Mathematical Physics.
Good books on ODE would be:
1) Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard
2) Differential Equations by George F. Simmons
3) Ordinary Differential Equations by E.Coddington
For PDE, I request you to take a look at this link: http://www.physicsforums.com/showthread.php?t=175382
I am a big fan of Bentley and Cooke's Linear Algebra with Differential Equations. It was published in the early 70's, it is black and white (i.e. not flashy), and has excellent references throughout the text (such as a footnote pointing to Kamke).
Furthermore, the topics of differential equations and linear algebra are introduced side by side at the introductory level. I think this is an ideal text for the beginning engineering student that holds him or herself to learning a bit more mathematics than their peers.
I am not sure if it would be considered a classic or not, but I absolutely love Differential Equations with Boundary-Value Problems by Dennis G. Zill and Michael R. Cullen. I feel as though the definitions are written and explained in such a way, that if one was not a mathematician, they could still grasp a basic understanding of many differential equations concepts. I am not familiar with many other elementary differential equations textbooks, but liked that this text goes into some details regarding Fourier series, integrals, and transforms, Legendre series, Bessel functions, and the Sturm-Louiville problem.
I still use this text as a reference for some of my graduate level ODE issues, as it is more comprehensible than most graduate level Diff-EQ texts. I also use it quite frequently when tutoring students in undergraduate level differential equations, as it contains a myriad of detailed examples and worked problems as well as a variety of real-world applications.
Though I'm not sure if it exactly meets your criteria, my favorite PDE book is Partial Differential Equations: An Introduction by Walter A. Strauss.