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Starting solutions corresponding to the oscillating motion of a Maxwell fluid between side walls perpendicular to a plate are established using integral transforms. Such solutions are scarce in the literature, the motion of the fluid being due to an oscillating shear on the boundary. The solutions corresponding to the motion over an infinite plate that applies an oscillating shear to the fluid are obtained as limiting cases of general solutions. All solutions are presented as a sum between steady-state and transient solutions and can easily be particularized to give the similar solutions for Newtonian fluids. They describe the motion of the fluid some time after its initiation. After that time, when the transients disappear, the motion of the fluid is described by the steady-state solutions which are periodic in time and independent of initial conditions. However, they satisfy the boundary conditions and governing equations. Finally, the distance between walls for which the velocity of the fluid in the middle of the channel in unaffected by their presence and the required time to reach the steady-state are graphically determined. [ABSTRACT FROM AUTHOR]