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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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up vote 1 down vote accepted

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

$$Y(a)=\lambda\left(k_2\atop k_1\right)\;.$$

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