<p>One of the most essential stages in improving heat transmission is to use hybrid nanofluids instead of nanofluids. As a result, the purpose of this study is to investigate the impacts of titanium oxide (TiO₂) and silver (Ag) with water (H₂O) as the base fluid, as well as the fluid flow and heat generated by permeably exponentially stretching or shrinking sheets of hybrid nanofluid. To reach the results of this study, the methodology adopted here reduces the governing partial differential equation transfer in terms of nonlinearly coupled ordinary differential equations utilizing similarity transformation. The nonlinear coupled ODEs are solved using the “bvp4c” solver in the MATLAB package. When compared to previously reported data, the current findings are consistent. The authors emphasize fluid flow and heat transfer characteristics for various values of velocity and thermal slip, as well as suction and heat generating factors, in this study. It is significant to emphasize that there is a dual solution for hybrid nanofluid flow for a specific range of stretching, shrinking, and suction parameters. The evaluation of stability is performed to determine which solutions are stable and which are not stable. Subsequently appears to be that the first solution is stable and physically achievable, whereas the second solution seems unstable. The results reveal that increasing the velocity slip parameter decreases the skin friction coefficient while increasing the local Nusselt number. The Nusselt number for both the first and second solutions drops as the thermal slip parameter is increased. In the first solution, as the silver volume fraction parameter increases, so do the skin-friction coefficient and the Nusselt number. When both the thermal slip parameter and the heat generation parameter are increased in the first solution, the Nusselt number decreases. When the stretching or shrinking parameter is raised, the skin friction coefficient falls and the Nusselt number increases. The impacts of many significant parameters on the velocity and temperature profiles are explored and graphically displayed. The investigation of flow and heat transfer across a permeable exponentially stretching or shrinking sheet with velocity slip and thermal slip conditions in Ag–TiO₂/water hybrid nanofluids is unique and novel. This paper could also provide dual solutions. The findings obtained may be utilized to understand the properties and applications of generalized slip in boundary layer flow. The computations were performed using the bvp4c MATLAB tool on the PARAM Shavak high-performance computing (HPC) system.</p>