Consider $Z(s)$ which is a solution to the following ODE: $$Z''-kZ=ilZ,$$ where $k,l$ are functions of $s$ and $i=\sqrt{-1}.$ Then define, $$T(s) = -Z(s)\int_{s}^{s_K}\frac{ds'}{Z^2(s')}.$$ Note that here we assume that $Z$ does not vanish at $s_K.$ Then the author of the paper where these expressions are mentioned states that, $T$ satisfies: $$T''-kT=ilT.$$ I am not sure if this is the case, since $$T' = -Z'\int_{s}^{s_K}\frac{ds'}{Z^2(s')}+\frac{1}{Z} = \frac{Z'T+1}{Z}$$ and so, $$T''= -Z''\int_{s}^{s_K}\frac{ds'}{Z^2(s')}+\frac{Z'}{Z^2} - \frac{1}{Z^2} =\frac{Z''T}{Z}+\frac{Z'-1}{Z^2} =T(k+il) + \frac{Z'-1}{Z^2}.$$
This is the most I could reduce, I still cannot eliminate $Z'.$ Any ideas will be much appreciated.