We consider first and second order linear dynamic equations on a time scale. Such equations contain as special cases differential equations, difference equations, $q$-difference equations, and others. Important properties of the exponential function for a time scale are presented, and we use them to derive solutions of first and second order linear dyamic equations with constant coefficients. Wronskians are used to study equations with non-constant coefficients. We consider the reduction of order method as well as the method of variation of constants for the nonhomogeneous case. Finally, we use the exponential function to present solutions of the Euler-Cauchy dynamic equation on a time scale.