The present study aims to investigate the transient behavior of thick-walled cylinders under dynamic internal pressure. Analytical solutions are possible only for simple time-dependent pressure functions. The solution procedure presented is general in the sense that the pressure applied may be an arbitrary continuous function of time, impulsive or given in a discrete form. The material considered is isotropic and heterogeneous with properties varying in the radial direction termed as Functionally Graded Material (FGM). Laplace transform method is used and the inversion into the time domain is performed using the modified Durbin's method. Verification of the numerical procedure is performed by comparing the results with those of an analytical solution available in the literature for a simple exponentially-varying pressure. The inhomogeneity constant in the material property model is shown to have a significant effect on the transient response.