  The Boeder differential equation is solved in this work over a wide range of
$\alpha$, yielding the probability density functions (PDF), that describe the
average orientations of rod-like macromolecules in a flowing liquid. The
quantity $\alpha$ is the ratio of the hydrodynamic shear rate to the rotational
diffusion coefficient. It characterises the coupling of the motion of the
macromolecules in the hydrodynamic flow to their thermal diffusion. Previous
analytical work is limited to approximate solutions for small values of
$\alpha$. Special analytical as well as numerical methods are developed in the
present work in order to calculate accurately the PDF for a range of $\alpha$
covering several orders of magnitude, $10^{-6} \le \alpha \le 10^{8}$. The
mathematical nature of the differential equation is revealed as a singular
perturbation problem when $\alpha$ becomes large. Scaling results are obtained
over the differential equation for $\alpha \ge 10^{3}$. Monte Carlo Brownian
simulations are also constructed and shown to agree with the numerical
solutions of the differential equation in the bulk of the flowing liquid, for
an extensive range of $\alpha$. This confirms the robustness of the developed
analytical and numerical methods.
