International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 1, Pages 143-145
doi:10.1155/S0161171279000144

Alternative integration procedure for scale-invariant ordinary differential equations

Abstract

For an ordinary differential equation invariant under a one-parameter group of scale transformations x→λx, y→λαy, y′→λα−1y′, y″→λα−2y″, 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=∫xy−1/αdx. This alternative integration may go through, as it does for the example equation y″=kxy−2y′, in cases for which the customary dependent and independent variables (x−αy) and (ℓnx) do not yield an analytically integrable transformed equation.