Vehicle platooning can be formulated as an optimal control problem and many solving paradigms, such as Pontryagin's maximum principle-based and dynamical programming methods, have been recently developed. However, these methods usually rely on solving a group of necessary conditions or Hamilton-Jacobi-Bellman (HJB) partial differential equations, which is hard to calculate. Besides, due to the heterogeneous dynamics of different vehicles in a mixed and complex platoon which comprises of not only connected autonomous vehicles (CAVs), but also human-driven vehicles (HDVs), it is also challenging to coordinate the behaviors of different vehicles in an unified control framework. Here we provide a Neural Mixed Platoon Control (NMPC) framework, a novel control design for mixed vehicle platooning based on a neural ordinary differential equation (NODE). We first formulate an optimal control model that incorporates the heterogeneous dynamics of a leading CAV and several following HDVs. We use a neural network to parameterize a state-feedback controller and join the neural controller and the mixed platooning dynamics into the NODE solver to create a closed-loop and learnable controlled system. The resulting system can learn optimal control inputs driving the mixed platoon to evolve from a given beginning condition to the target state within a finite duration in an unsupervised manner. Finally, simulation results validate our suggested method's usefulness in terms of space headway and velocity tracking.