International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 1, Pages 143-145
doi:10.1155/S0161171279000144
Research notes
Alternative integration procedure for scale-invariant ordinary differential equations
Department of Physics, Drexel University, Philadelphia 19104, Pennsylvania, USA
Received 28 December 1978
Copyright © 1979 Gerald Rosen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.