This paper proposes a method to learn approximations of missing Ordinary Differential Equations (ODEs) and states in physiological models where knowledge of the system's relevant states and dynamics is incomplete. The proposed method augments known ODEs with neural networks (NN), then trains the hybrid ODE-NN model on a subset of available physiological measurements (i.e., states) to learn the NN parameters that approximate the unknown ODEs. Thus, this method can model an approximation of the original partially specified system subject to the constraints of known biophysics. This method also addresses the challenge of jointly estimating physiological states, NN parameters, and unknown initial conditions during training using recursive Bayesian estimation. We validate this method using two simulated physiological systems, where subsets of the ODEs are assumed to be unknown during the training and test processes. The proposed method almost perfectly tracks the ground truth in the case of a single missing ODE and state and performs well in other cases where more ODEs and states are missing. This performance is robust to input signal perturbations and noisy measurements. A critical advantage of the proposed hybrid methodology over purely data-driven methods is the incorporation of the ODE structure in the model, which allows one to infer unobserved physiological states. The ability to flexibly approximate missing or inaccurate components in ODE models improves a significant modeling bottleneck without sacrificing interpretability.