Analytical Solution of the Proprio-Graviceptive equation for shoot gravitropism of plants

We derive the analytical solutions to the second order generalised gravi-proprioceptive equation given in our recent paper [Bastien et al. 2012]. These equations show how plants adjust to the surrounding gravitation field and highlight the fact that the plant must be able to not only sense its local posture with respect to the gravitational field, but also to sense its own local curvature. In [Bastien et al. 2012] we obtained explicit analytical solutions of these equations in terms of (sums of) Bessel functions, and in the present paper we derive these solutions.