Alternative integration procedure for scale-invariant ordinary differential equations

For an ordinary differential equation invariant under a one-parameter group of scale transformations x → λ x , y → λ α y , y ′ → λ α − 1 y ′ , y ″ → λ α − 2 y ″ , 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 ξ = ∫ x y − 1 / α d x . This alternative integration may go through, as it does for the example equation y ″ = k x y − 2 y ′ , in cases for which the customary dependent and independent variables ( x − α y ) and ( ℓ n x ) do not yield an analytically integrable transformed equation.