Flow-induced vibration is a common phenomenon in fluid dynamics, finding applications from aeroelasticity to blood flow in arteries. This study presents a finite element formulation as well as Galerkn's method for analysing the free and forced vibration of pipes that carry either harmonically pulsating or uniform flows. Relevant finite element matrices and coefficients of Galerkin's method are derived to address dynamic and stability issues. Notably, the damping and stiffness matrices are shown to vary with time. Galerkin's approach is also employed to validate the results obtained through the finite element method. Furthermore, this research explores the dynamic and stable behaviour of pipes with various boundary conditions, such as simply supported, clamped, propped cantilever and cantilever pipes resting on elastic foundations. Additionally, it addresses the problems related to axially functionally graded tapered pipes conveying either Newtonian or non-Newtonian flows. The Connors critical velocity is also discussed as a measure of initiating fluid-induced instability. It is observed that the natural frequency and critical velocity increase with the flow index, Winkler foundation stiffness and mass ratio. Several numerical examples are provided to validate the applied methodologies. Overall, this research contributes valuable insights into the complex interactions between flow and vibration in various pipe configurations with practical implications across multiple engineering disciplines.