Here's how I broke it down, it has been a little while since I've been in a math class * Differentiable to all orders means that for each derivation, no cusp will appear in the curve. A cusp means that the next order of derivation will not be defined at that point on the curve.

* 'The symmetry is a smooth invertible transformation mapping solutions of the X to solutions of the Y'. - I now understand that the stuff I just paraphrased means that it's just a mapping, and that it's invertible. - ODE = Ordinary Differential Equation. Cool. Rings a bell. It looks like ^ODE^ is just the next order of derivation? And this mapping, the symmetry, is just describing how the next order of derivation relates to the first (I think, that is not exactly clear in the time I spent).

* Invertible means the Jacobian is nonzero... Describing to a sophomore that a mapping is invertible in these terms is pretty vague (this section is supposed to be accesible to sophomores). The Jacobian is the determinant of a particular form of matrix, http://mathworld.wolfram.com/Jacobian.html

So aside from that last bit it came apart okay. I have noticed that when you have completed a certain amount of math (or any topic) it is hard to exclude certain bits or to describe things in a simpler fashion