I have used Kreyszig, Advanced Engineering Mathematics, fifth edition, as an undergrad, a practicing engineer, and in grad school. I love it.
(1) It is in English, composed in English, good English, but written by a German native speaker mathematician. Not sure that this matters but might be positive.
(2) One of high repute, check out his books on differential geometry or functional analysis.
(3) It is written by a mathematician and better than that, a math TEACHER. So while he is (was, as he is dead) capable of hard core math like above, he has a good feel for EE students, for writing enough text but not too much, for good problems (not just a few hard ones, but lots of different hardness, and plenty that just allow trying the new concepts to get familiarity).
(4) Real mathematician as opposed to practitioner. This can be advantageous over math books written by physicists or engineers. (They tend to be more confusing...and it is easier to cover some of the math within physics from that physics slant, but then to get it from a math slant in math class...makes things click to see it from both sides: vector calc in Wangness E&M versus 3rd semester calc class, kinematics in college physics versus derivates/integrals in calc 1/2 classes.) The one example of a practioner book that I own is Arfken Weber for physics: has some overlap, but a little more advanced...but very spotty and poorly written...read the negative reviews on Amazon.
(5) While written by a mathematician, it is a mathematician with both an familiarity of engineering/science and a sympathy to engineers and scientists. This can be in contrast to many mathematicians and math teachers who lack strong general engineering/science education or sympathy.
(6) coverage includes:
a. ODE: basic, efficient, coverage in five chapters, 250 pages (excepting systems covered in LA part and numeric methods covered in NM part (see below). All the topics in longer books (first and second order ODEs, series method, Bessel functions, Sturm Louisville, phase plane, Laplace Transform, etc.), but more efficiently (IMO). Most reviewers are high on this part of the book as just well done. [Note however, that in the US, it is common for engineering students to cover this with a specialized text and dedicated ODE course. But it is in there.
b. Linear algebra: two chapters, 100 pages. Efficient, engineer's coverage of matrices and key operations. In my experience, many engineers and physicists get LA only within their "math methods" or "engine math" course. But some of course have dedicated courses (as with all the topics).
c. Vector calculus: Two chapters, 100 pages. Good, efficient coverage. Note that in the US, this is normally covered in 3rd semester calculus, usually with a book that covers calc 1/2/3 (Thomas, Swokowski, etc.). However, K has it as a topic on it so it matches your need. But I had covered ODE and VC previous to this book.
d. Fourier series and PDEs. A chapter on each, 100 pages total. K covers F series first, which I quite like as there are a lot of applications BEFORE you get to PDEs. And it just seems a little easier to cover it this way straight up versus as an evolution of harder PDE problems. I think this is nice for EEs also who use FS a lot (as the en passent coverage in typical PDE book may not be optimal.) The PDE chapter covers all the major topics (heat equation, circular membrane, blabla). However it is just a single chapter with 13 sections. I think it is good for an engineers treatment and as part of a math methods course. But as with any of these topics, you can say maybe you should get a book on each topic instead. That said, I find the PDE section very straight forward. Not too applied (PDEs can be confusing because of the strong emphasis on applications...and kids will see that slant in their thermo classes regardless.) But also not too rigorous as to be annoying. Bessel functions are revisited within PDE chapter.
e. Complex analysis: Five chapters, 200 pages: mathematical but an emphasis on topics of interest to engineers (mapping, contour, residue blabla, etc....and derivations as motivation but not proof as the main emphasis.) Series are covered strongly in a chapter here as well. There is some "real analysis" (like the series) but more useful to engineers "advanced calculus". Not annoying topology proof stuff.
f. Probability and statics: One chapter, 100 pages. I didn't use this section, had a separate class for it. I think a lot of math methods courses would not use this section. But it's in there. Looks straightforward and clean and fine.
g. Numerical methods and optimization: Two chapters 60 pages. Good section on different methods and error analysis. LP and simplex method in optimization. Looks fine. Never used it.
Coverage note: First and second semester calculus were NOT covered AND considered a prereq. Epsilon-delta was not covered either. In the US, epsilon-delta is typically covered (quickly) within normal calc course...so you get an exposure...but then you move to various manipulations and applications (partial fractions integration, min/max problems etc.) I would also add that I never felt my lack of MORE epsilon-delta awareness stopped me from doing hard calculus problems through EE, physics (even some grad courses). I bet Feynman felt the same way. (I bet Euler did too!) Now Bessel functions is a different story...those darned Yo and Jo things are everywhere.
It is also normal in the US for the first year calc course to have (towards course end) a short ODE section and a short sequence/series coverage. [This is helpful for students who stop here with math. However, engineers will normally get a full ODE course as a requirement later. Deeper knowledge of series divergence and the like may not be a required course.] But K doesn't require any carryover of this "baby ODE" or "baby series" knowledge (nor do other books in general in the US...that just ends up being review for the kids.)
The book does not cover calculus of variations or group theory. (Arfken and Weber do, poorly.) FWIW, I don't think these are high priority topics for typical EE undergrad, but FYI. Of course some specialized topics like signal processing or Boolean algebra are also not covered. I find most EEs do fine with how that is covered in the EE texts themselves.
Length note: this book is about 4-6 semesters depending on student strength. So picking and choosing becomes a need for shorter courses. My one semester class covered LA, small part of F series and large part of PDEs. A two semester class for physicists would add the complex analysis chapters. It is a good reference though.
Modeling: Preface talks a bunch about emphasis on modeling and interpretation. I am really not in favor of so much of that in a math book (you get it in thermo and controls class regardless). But it didn't end up bugging me. So either he did it in a non-annoying manner or he just wrote about it a bunch in the preface to keep engineer professors happy.
Layout note: I found the book to be very well printed, few mistakes (I only know of one typo), and well laid out...text appropriate amount/tone (not too formal, not too chatty). Enough examples for engineers. Nice layout with learning objectives and chapter summaries and nice font usage. No Fraktur (darned Altschrift)...which is good...I hate that. The Greek letters "squiggle" and "other squiggle" may have turned up a teensy amount but not too much and not ostentatiously (like when a gamma would work fine and be normal engineering usage). There were also very short historical footnotes on major mathematicians (nationality and birth/death). I thought it was positive as there are a lot of new names (Legendre and the like) that come in and some other reference for them is helpful. But it was just tiny notes and didn't get in the way of the math.
Size: It has a nice heft to it and can be used to bash ganglions down. In this it is similar to Arfken and Weber or Brealey and Myers. ;-)
Answers: 5th edition had half the answers. I hate books that don't have the answers. So it is batting 50% on that, with me. But it is better than some books like A/W that had minimal answers. You should check more recent vintage though. There may be some accompanying answer manuals out there but you would have to check out the coverage (is an annoying commercial practice of US publishers of having "student solution manuals" that only have half the answers...this is driven by commercial aspect of professors preferring it this way...is an annoying way that we have become LESS LIBERAL than c. 1900 math books.)
Availability: It is probably the "Coke" of US Engine Math books so easily available. I would go for an earlier edition (used) if price is an issue.