I think a number of people have been in a very similar situation to you, in a course on differential equations or otherwise, and have looked for an alternate source for the material. I myself am very familiar with the problem.
There are a number of excellent textbooks on the subject that sell for less that $15, my personal favorite of which is Tenenbaum/Pollard, but this opinion is obviously not objective and you, along with a number of other answeres seem to find significant faults with their development of the subject.
I will list some other possible inexpensive or free resources that I am at least mildly familiar with:
- Agnew's Differential Equations is an old book that treats the subject very classically in a way similar to Tenenbaum/Pollard. One of the greatest aspecsts of this book is its index, which is quite extensive. It relies heavily on physical applications. And you can get it for less than \$5 from Amazon including shipping. This is not a particularly famous choice, as I purchased a copy at a local used bookstore for \$1 but it is nonetheless excellent.
- MIT OCW 18.03 course, as others have pointed out, is a complete set of lectures, notes, and problem sets that would basically make up a course if you were to take from a spectacular lecturer. I would not reccommend buying the textbook suggested on the syllabus page, however. The supplemental notes from what I remember are excellent.
- Paul's Math Notes have a set of lecture notes on differential equations that cover all of the topics you are asking for. I have never read these, although I've seen them refrenced quite frequently and they are considered excellent from what I have seen.
- Google site:.edu filetype:.pdf. Google is an incredible tool, and is far more extensive that most people imagine. They have a number of operators that refine your search, and their engine is so powerful that there is an entire area of computer security devoted to using google to hack websites. By using the operators site:edu and filetype:pdf we restrict our search to .pdf files from academic institutions. By selecting a query such as "Bernoulli equations" with the operators described (i.e., type
"Bernoulli equations" site:edu filetype:pdf into the google search bar) you will recieve a plethora of lecture notes and descriptions of whatever topic you are looking for. Read one, read 5, read 1000. By reading the lecture notes of many different lecturers you can grasp a topic from a number of different viewpoints and methods of development simultaneously, and this provides an excellent supplement to your course and or any of the resources I described above.
I wish you luck with your course. Have fun learning.