I have recently discovered the fascinating area of elastica and have been reading Raph Levien's work on their history (https://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.pdf). I understand that the general problem is to minimise the integral of the squared curvature of a curve, which is referred to as the 'bending energy' in many sources:
$$ E[\kappa(s)] = \int \kappa(s)^2 ds. $$
Minimising the integral leads to Equation 12 in Levien's work:
$$\frac{dy}{dx} = \frac{a^2 - c^2 + x^2}{\sqrt{(c^2 - x^2)(2a^2 - c^2 + x^2)}}.$$
Levien then proceeds to describe Euler's discovery of the nine classes of elastica, based upon combinations of $a$ and $c$ in the above equation. However, the meaning of these parameters is not explained and the reference Levien gives is inaccessible at my current knowledge. My question is, if I think of two fixed points and their first derivative conditions (perhaps also a length constraint?), how do I use the second equation to determine the form of the curve joining these two points, or even which of the nine classes of elastica my hypothetical curve belongs to? For example, if I have $ P_1, P_2 = (0,0), (3,3) $, with derivative conditions $P^{'}_1 = 1$ and $P^{'}_2 = 1$, I know the answer should be a straight line belonging to the first class of elastica, but how does that fall out of Equation 2?