Alternative integration procedure for scale-invariant ordinary differential equations

For an ordinary differential equation invariant under a one-parameter group of scale transformations x ￠ ’ x, y ￠ ’ ±y, y ￠ € 2 ￠ ’ ± ￠ ’1y ￠ € 2, y ￠ € 3 ￠ ’ ± ￠ ’2y ￠ € 3, etc., it is shown by example that an explicit analytical general solution may be obtainable in parametric form in terms of the scale-invariant variable = ￠ xy ￠ ’1/ ±dx. This alternative integration may go through, as it does for the example equation y ￠ € 3=kxy ￠ ’2y ￠ € 2, in cases for which the customary dependent and independent variables (x ￠ ’ ±y) and ( ￠ “nx) do not yield an analytically integrable transformed equation.