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My book makes a statement about the symmetry of the family of solutions to this diff eq I don't quite understand.

"Symmetry. If we replace x with -x on both sides of 1.17, the right hand side is unchanged but the left hand side changes signs. So the family of solutions is not symmetric across the y-axis. However, if we simultaneously replace x with -x and y with -y, then we obtain 1.17 back again. So the family of solutions of 1.17 is unchanged under simultaneous interchange of x with -x and y with -y. This means that the family of solutions is symmetric about the origin."

It doesn't showcase the algebra so that adds some more difficulty for me.

I guess my biggest question is this: I can see how facts about the symmetry of the ODE can suggest properties of the solution curves. In this case, though, how does the above example show anything about the symmetry of the solution? Isn't manipulating the ODE in this way showing symmetry for the ODE itself? That last point wouldn't make sense either, because this ODE is symmetric about the y-axis.

Thoughts?