We calculate the response time probability density for two processor-sharing (PS) queues in tandem for the first time. We use the generating functions method to obtain a system of functional equations for the Laplace-Stieltjes transform (LST) of the conditional response time distribution, where the conditioning is on the queue lengths at the time of arrival of the tagged task. Similar equations have been derived previously; however, they remained unsolved. Here, we utilise a numerical approximation technique to find the solution for the conditional LST, whereupon the unconditional LST is obtained directly using the product-form solution of the queue length probabilities. We invert the LST numerically to obtain the response time density itself. Furthermore, an efficient, recursive formula is obtained for the calculation of response time moments. The results are compared against regenerative simulation and show remarkable accuracy. Finally, the PS-PS tandem network is compared against the PS-FCFS and FCFS-FCFS systems with the same parameters.