The thermoelasticity problem in a thick-walled isotropic and homogeneous cylinder is solved analytically using finite Hankel transform and Laplace transform. Time-dependent thermal and mechanical boundary conditions are prescribed on the inner and the outer surfaces of the hollow cylinder. For the thermal boundary conditions, the temperature itself is prescribed on the boundaries. For the mechanical boundary conditions, the tractions are prescribed on both the inner and the outer surfaces of the hollow cylinder. Obtaining the distribution of the temperature throughout the cylinder, the dynamical structural problem is solved and closed-form relations are derived for radial displacement, radial stress and hoop stress. As a case study, exponentially decaying temperature with respect to time is prescribed on the inner surface and the temperature of the outer surface is considered to be zero, where the mechanical tractions on the inner and the outer surfaces of the hollow cylinder are assumed to be zero. On solving the dynamical thermoelasticity problem, a thermal shock was observed after plotting the results. Using the obtained plots, instants of reaching dilatation wave to specific radial positions are computed and compared with those from the classical formula.