We introduce the Schwarzian, simplified Schwarzian, and Schwarzian Korteweg-de Vries equations on a time scale that are invariant under the fractional linear transformations. As an application, we derive their solutions and establish the invariant disconjugacy condition for second order dynamic equations on a time scale. Furthermore, we consider the Ermakov dynamic equation and the Ermakov-Lewis adiabatic invariant on a time scale. We discuss specific examples of discrete, quantum, and continuous time scales to compare our equations with the well-known ones. We also derive the linearization of the Ermakov equation and the corresponding Painleve equation on a time scale.
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