Pulse-coupled oscillators (PCOs) are used as models for oscillatory systems in diverse fields such as biology, physics, and engineering. When correctly coupled, PCOs can display sophisticated emergent dynamics for large numbers of oscillators. Here, we propose an algorithm and hardware implementation of PCOs to emulate arbitrary systems of linear differential equations (DEs) with inputs, which are similar to the equations used in feedback control laws or linearizations of nonlinear systems. We show that m populations of oscillators can solve a set of m-dimensional linear DEs with simple coupling schemes, and crucially, without the matrix multiplications required in Euler integration. The emergence of linear dynamical systems in networks of PCOs occurs when the number of oscillators within a population becomes large through an analytically exact mean-field derivation. In addition, a hardware architecture of PCOs for digital implementation is proposed and realized on an ultralow power FPGA as a proof of concept. These results show that there are simple coupling schemes for networks of pulse-coupled oscillators that collectively compute complex dynamical systems. These PCO networks also have an immediate implementation as low power neuromorphic edge devices.