Symmetries, first integrals and phase planes of a third-order ordinary differential equation from thin film flow

The third-order ODE yny‴=1yny‴=1 obtained by investigating travelling-wave solutions or steady-state solutions of the lubrication equation is considered. The third-order ODE yny‴=1yny‴=1 admits two generators of Lie point symmetries. These generators of Lie point symmetries effect a reduction of the third-order ODE to first order. The problem is to determine the initial values of the second derivative, when the initial height and gradient are specified, for which a solution to yny‴=1yny‴=1 touches the contact line y=0y=0. Phase planes corresponding to different representations of the first-order ODE for the cases n<2n<2, n=2n=2 and n>2n>2 are analyzed. For the case n<2n<2 we are able to determine the initial values of the second derivative for which the solution touches the contact line. For n≥2n≥2 no values of the initial second derivative are obtained for which a solution touches the contact line. A symmetry reduction of autonomous first integrals of the third-order ODE yny‴=1yny‴=1 is then investigated. For the cases n=0n=0, n=5/4n=5/4 and n=5/2n=5/2 the third-order ODE admits second-order autonomous first integrals. The case n=5/4n=5/4 is special because the second-order autonomous first integral admits the same two generators of Lie point symmetries as the original third-order ODE and can hence be reduced to an algebraic equation. Investigations of the phase plane for the case n=5/4n=5/4 shows that the original third-order ODE satisfies the contact line condition y=0y=0 for initial values of the second derivative y″(0)≤−3.