I apologise at the start if this question looks incomprehensible. I have transformation of space curves in $\mathbb{R}^3$ of the form $$ y = x + \delta, $$ where $$\delta =\dfrac{d}{ds_t}\Bigg[c+\dfrac{d^2c}{ds^2_b}\Bigg]T- \dfrac{d^2c}{ds_b^2}N + \dfrac{dc}{ds_b}B$$
Here $ds_t = \kappa ds$ and $ds_b = \tau ds$, $\kappa,\tau$ being the curvature and torsion of $x$ curve(parametrised w.r.t arc-length $s$) and $c$ is function chosen such that $$\dfrac{ds_y}{ds} = \Bigg[1 + \kappa f(c)\Bigg]$$ and $f(c) = \dfrac{d^2}{ds_t^2}\Bigg[c+\dfrac{d^2c}{ds^2_b}\Bigg] + \dfrac{d^2c}{ds_b^2}$.
I am interested in in determining $c$ for the particular case where $$f(c) = \dfrac{(\tau^2 +\kappa^2) -\kappa}{\kappa^2}.$$
I am not sure how to proceed with this problem. The paper I am referring to approaches it by setting $$f(c)= \dfrac{\tau^2}{\kappa^2}\Gamma \Big(\dfrac{dc}{ds_b} \Big)$$ and says that $\Gamma(B)= 0$ would be the differential equation obtained if one eliminate $T$ and $N$ from the Frenet Serret formulae. In such a case, one obtains the solution as $$\dfrac{dc}{ds_b} = r\Big<u, B \Big> - r\Big<B, \int K dx\Big>,$$ where $K = \kappa f(c)$, $r$ being a scalar constant and $u$ a vector constant and so the transformation is given as $$ y = x +\int K dx -ru$$ Can anyone help me understand how the solution was arrived at? Or is there a better way of solving this ODE??
Here is a link to the paper I referred.