Linear and Riccati equations in generating functions for stellar models in general relativity

It is shown that the expressions for the tangential pressure, the anisotropy factor and the radial pressure in the Einstein equations may serve as generating functions for stellar models. The latter can incorporate an equation of state when the expression for the energy density is also used. Other generating functions are based on the condition for the existence of conformal motion (conformal flatness in particular) and the Karmarkar condition for embedding class one metrics. In all these cases, the equations are linear first-order differential equations for one of the metric components and Riccati equations for the other. The latter may be always transformed into second-order homogenous linear differential equations. These conclusions are illustrated by numerous particular examples from the study of stellar models.