The analytic solutions obtained through the finite Fourier sine or cosine transforms or the Laplace transform for two different motions of Newtonian fluids between two horizontal infinite parallel plates are different as form but equivalent. In addition, the solutions obtained by means of the Laplace transform, which are new in the existing literature, are established in simple and elegant forms in terms of the exponential function and the complementary error function of Gauss which is a tabulated function. © 2020, Springer Nature India Private Limited.