INVARIANT SOLUTIONS FOR ORDINARY DIFFERENTIAL-EQUATIONS

An invariant solution of a differential equation is a solution of the differential equation which is also an invariant curve (surface) of a group admitted by the differential equation. For an ordinary differential equation (ODE) such solutions can be found without determining its general solution. A theorem is proved which shows that for an ODE invariant solutions can be found by solving an algebraic equation derived from the given ODE and the infinitesimals of an admitted Lie group of transformations. For first-order ODEs it is shown that separatrices and envelope solutions are invariant solutions for any admitted Lie group. Several examples are given.