Is it cheating to use solutions to identical problems from other universities to do my homework?
I do not view it as generally unethical to refer to these solutions.  This situation is more complex than I think some other answers have admitted. Here is a list of claims: 


In an ideal world, the point of homework is for the student to learn the material.
In a perfectly ideal world, we would not need to grade homework, because students would do it on their own to master the material. They might refer to other people's solutions to see if theirs are correct, and that would be fine.  
Experience shows this world is not perfect. Students will often skip ungraded homework, and their learning and exam grades will suffer.
So instructors assign homework for a grade. But this isn't because the grade is really important: it's because we want the students to do the homework and learn the material!
Some students then get the idea that the grade is the real goal of the homework, and simply copy their assignment from others. Professors often find this unacceptable.  


One important point that others have answered is that, if you are going to turn in the homework, what you turn in should reflect your own understanding of the assignment. But, equally importantly, it is important to let yourself struggle with problems for a while before looking up the answer.  That is the only way to really learn how to solve problems. 

Most professors accept that the internet exists - we know you can look up other people's answers. It used to be that fraternities had giant files of old homework and exam answers for this purpose (maybe they still do).  And students study in groups all the time - research shows study groups can dramatically increase learning.  So getting help is not a bad thing. 

But you don't want to get help too quickly. Make a genuine effort to answer the problems yourself first. If you find that you are looking up the answers to all the problems (even the easiest ones), then something is off - try going for more tutoring, or studying more before doing the homework.

If you find that you occasionally need to look up one of the most difficult problems, that's perfectly normal (but it still wouldn't excuse directly copying the solution into your homework, of course). 

Of course, the usual caveats apply: some professors may specifically tell you not to collaborate with anyone or use any other resources. But most professors know that students usually collaborate with each other on homework (e.g. study groups) and know that students can look up answers using other resources. We have no problem with that, as long as each student's submission reflects their own understanding in the end.