A quite serious error in Mathematica 7 in my opinion is that it thinks $ \sqrt{x^2} =x$, not $|x|$, leading for example to 2 solutions to the following differential equation: $$ y'(x) = 2 y(x) (x \sqrt{y(x)} - 1) \quad y(0) =1$$ Mathematica happily gives the following solutions: $$ y(x) \rightarrow \frac{1}{(1-2 e^x +x)^2}, \quad y(x) \rightarrow \frac{1}{(1+x)^2} $$ Of course, it is a theorem that there is a unique solution to a differential equation of this type, but that doesn't mean my students hand in the wrong answer in droves...

Mathematica code: FullSimplify[DSolve[{y'[x] == 2 y[x] (x Sqrt[y[x]] - 1), y[0] == 1}, y[x], x]]