  Ordinary differential equations (ODEs) are used to model dynamic systems
appearing in engineering, physics, biomedical sciences and many other fields.
These equations contain unknown parameters, say $\theta$ of physical
significance which have to be estimated from the noisy data. Often there is no
closed form analytic solution of the equations and hence we cannot use the
usual non-linear least squares technique to estimate the unknown parameters.
There is a two step approach to solve this problem, where the first step
involves fitting the data nonparametrically. In the second step the parameter
is estimated by minimizing the distance between the nonparametrically estimated
derivative and the derivative suggested by the system of ODEs. The statistical
aspects of this approach have been studied under the frequentist framework. We
consider this two step estimation under the Bayesian framework. The response
variable is allowed to be multidimensional and the true mean function of it is
not assumed to be in the model. We induce a prior on the regression function
using a random series based on the B-spline basis functions. We establish the
Bernstein-von Mises theorem for the posterior distribution of the parameter of
interest. Interestingly, even though the posterior distribution of the
regression function based on splines converges at a rate slower than
$n^{-1/2}$, the parameter vector $\theta$ is nevertheless estimated at
$n^{-1/2}$ rate.
