  We consider population modelling using parametrised ordinary differential
equation initial value problems (ODE-IVPs). For each individual drawn randomly
from the unknown population distribution, the corresponding parameters for the
ODE-IVP cannot be measured directly, but a vector of covariates is given, and
one component of the solution to the corresponding ODE-IVP is observed at a
fixed finite time grid. The task is to identify a covariate-parameter
relationship that maps covariate vectors to parameter vectors. Such settings
and problems arise in pharmacokinetics, where the observations are blood drug
concentrations, the covariates are clinically observable quantities, and the
covariate-parameter relationship is used for personalised drug dosing. For
linear homogeneous ODE-IVPs with vector fields defined by matrices that are
diagonalisable over $\mathbb{R}$, and for fixed time and random covariate
design, we use recent results of Nickl et al. for Bayesian nonlinear
statistical inverse problems, to prove posterior contraction and Bernstein--von
Mises results for the unknown covariate-parameter relationship. We analytically
demonstrate our results on an example from the pharmacokinetics literature.
