Skimmed a lot of this textbook。I hesitate to say this， but this may be suitable as a reference book for elementary DEs。 The theory is exposited here， but of course it doesn't teach you as much as， say， Arnol'd's ODE （for example）。 But everything is more or less rigorously proved （definitely rigorous enough without going into all the analysis stuff）， and everything is generalised to the the n-dimensional space and complex number field。If you want to figure out how to solve an equation， go for thi Skimmed a lot of this textbook。I hesitate to say this， but this may be suitable as a reference book for elementary DEs。 The theory is exposited here， but of course it doesn't teach you as much as， say， Arnol'd's ODE （for example）。 But everything is more or less rigorously proved （definitely rigorous enough without going into all the analysis stuff）， and everything is generalised to the the n-dimensional space and complex number field。If you want to figure out how to solve an equation， go for this book （rather than Boyce and DiPrima's horrible book）， which seems to be popular in universities in the USA）。 Also better than Paul's online maths notes because there's a lot of guesswork in the solving of DEs there。 Elsgolts shows that it's actually not that ad-hoc。No comment on the Calculus on Variations part。Plenty of examples。 Typesetting not that great， but it's free from the respectable Soviet-era Mir Publishers （and a bit old）。 。。。more