Identities for homogeneous utility functions

Using a homogeneous and continuous utility function to represent a household's preferences, we show explicit algebraic ways to go from the indirect utility function to the expenditure function and from the Marshallian demand to the Hicksian demand and vice versa, without the need of any other function. This greatly simplifies the integrability problem, avoiding the use of differential equations. In order to get this result, we prove explicit identities between most of the different objects that arise from the utility maximization and the expenditure minimization problems.