I have a second-order (Sturm-Liouville) differential equation for $u$ and want to express the boundary condition at (say) a, $k_1u(a)-k_2u'(a)=0$, in the form Y(a) = C where Y and C are column vectors and $Y=[y, y']^T$. I would appreciate your advice on the best method to do this, providing it is possible.
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I'll assume that you meant $Y=[u, u']^T$ (since you didn't introduce $y$ and $y'$).
Note that you can write your boundary condition as a dot product:
$$\left(k_1\atop -k_2\right) \cdot Y(a)=0\;.$$
So $Y(a)$ must lie on the line orthogonal to $\left(k_1\atop -k_2\right)$, and that line can be parametrized as