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What do I know as regards lie groups?

Solving separable equations somehow exploits the fact that the constant of integration [tex] C \ = \ y \ - \ \int f(x) dx[/tex] is a one-parameter group mapping solutions into solutions, & further that the method of change of variables is (apparently?) nothing more than a method of finding a coordinate system in which a one-parameter group of translations/rotations/...??? is admitted so that separation of variables is possible (not sure if that's only what SoV is good for, that just seems to be the implication!).

Solving Euler-Homogeneous equations somehow exploits the fact that the differential equation y' = f(y/x) admits a group of scalings, T(x,y) = (ax,ay), as in this link (bottom of page 23), thus because of this one can use Lie theory to solve these equations as well.

What am I asking for?:

I've tried to teach myself this material a while ago & failed, built up a bit of a mental block when trying again & failed again, went of asking grad students & professors who hadn't come across much of this material & so am now here with another attempt, basically all I need is for someone to explain what's going on with lie groups in general in light of what I've said I know about them & to kind of give the intuition behind what the general process is, how powerful it is etc... I was thinking maybe along the lines of the first chapter of this book, but whatever you think really, would just be good to have someone to ask questions of who knows this stuff!