Further transformation properties of generalised inhomogeneous nonlinear diffusion equations with variable coefficients

We consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form xput=[xqunux]x. We present further transformation properties for this nonlinear class of equations that do not appear in the literature. In particular, we map this class of variable-coefficient into constant-coefficient evolution equations. We also introduce hodograph and generalised hodograph transformation. For a specific form of this class we derive an auto-hodograph transformation. New nonlocal symmetries known as potential symmetries are found. Finally, examples of exact solutions are given.