Symmetries, first integrals and phase planes of a third-order ordinary differential equation from thin film flow (Q2389888)

The author investigates the third-order ODE \[ y^ny'''=1 \] subject to \[ y(0)=1,\;y'(0)=0\text{ and }y''(0)=\lambda<0, \] where \(n\) is a positive rational number and the constant \(\lambda\) is to be determined so that the solution satisfies a certain geometrical condition. This problem admits two generators of Lie point symmetries and therefore can be reduced to a first-order OPD. Phase planes for different values of the parameter \(n\) are analyzed.