The existence of correlation among mental disorders is a well-known phenomenon. Multi-task learning (MTL) has been reported to yield enhanced detection performance of a targeted mental disorder by leveraging its correlation with other related mental disorders, mainly in textual and visual modalities. The validation of the same on audio modality is yet to be explored. In this study, we explore homogeneous and heterogeneous MTL paradigms for detecting two correlated mental disorders, namely major depressive disorder (MDD) and post-traumatic stress disorder (PTSD), on a publicly available audio dataset. The detection of both disorders is interchangeably employed as an auxiliary task when the other is the main task. In addition, a few other tasks are employed as auxiliary tasks. The results show that both MTL paradigms, implemented using two considered deep-learning models, outperformed the corresponding single-task learning (STL). The best relative improvement in the detection performance of MDD and PTSD is found to be 29.9% and 28.8%, respectively. Furthermore, we analyzed the cross-corpus generalization of MTL using two distinct datasets that involve MDD/PTSD instances. The results indicate that the generalizability of MTL is significantly superior to that of STL. The best relative increment in the cross-corpus generalization performance of MDD and PTSD detection is found to be 25.0% and 56.5%, respectively.