Anyway, when couching the 'DSolve' command as follows: DSolve[y'[x] == (ArcSin[x]/(1 - x^2)^(1/2))/y[x]^2), y[x], x]... ...I get the empty set symbol (ie: {{ }} ...)

It works for me. You have an extra ) in there after y[x]^2. So I do not know how you got an answer instead of a syntax error. May be some copy/paste error. Any way, here is on version 10.02

Clear[y, x];
DSolve[y'[x] == (ArcSin[x]/(1 - x^2)^(1/2))/y[x]^2, y[x], x]

gives

{{y[x] -> -(-(3/2))^(1/3) (ArcSin[x]^2 + 2 C[1])^(1/3)}, {y[x] -> (3/2)^(1/3) (ArcSin[x]^2 + 2 C[1])^(1/3)}, 
  {y[x] -> (-1)^(  2/3) (3/2)^(1/3) (ArcSin[x]^2 + 2 C[1])^(1/3)}}

Not sure if this is helpful, but I thought I'd post it JIC. I was working on my laptop today (I have one of your 'prime' or 'premium' accounts, so I can run your product on my main work-oriented computer, and also on my laptop). The problem I'd encountered was on my desktop, but since I was there at the laptopI tried your input. Worked perfectly. Tonight, just out of compulsion, when I got back to my desktop I tried it on the same (still open) notebook that I've been playing with for the last 36 hours. Interestingly, I got the 'null set' result? So, being the suspicious geek that I am, I closed that notebook, shut Mathematica down, restarted and tried again. Works normally, gives right answer. So there was something broken in the notebook or the kernel.

I'm telling you this only in case this little vignette helps w/ some sort of potential instability in long-open windows that you guys may be working on. Thanks again for helping me!