Green's functions issue

[member 428835]

Hi PF!

The ODE $$g''(x) + (1-k^2)g(x) = f(x)\\ g(0) = y(\pi/3) = 0$$
where $f(x)$ is a forcing function and $k \in \mathbb N$ is a constant has a Green's function via variation of parameters as
$$
G_L = \frac{L(y)R(x)}{W} : 0<x<y<\pi/3\\
G_R = \frac{L(x)R(y)}{W} : 0<y<x<\pi/3
$$

with solutions $$L(x) = \frac{\sin(\beta x)}{\beta}\\
R(x) = \sin\beta(x-\pi/3)\\
W = \sin (\beta \pi/3)\\
\beta = \sqrt{1-k^2}$$

where $L(x)$ is the left sided solution and $R(x)$ is the right sided solution (obviously). I can verify this Green's function is correct for the given ODE. However, despite being correct, the inequalities look wrong. Why isn't the Green's function instead defined as
$$
G_L = \frac{L(y)R(x)}{W} : 0<y<x<\pi/3\\
G_R = \frac{L(x)R(y)}{W} : 0<x<y<\pi/3
$$
since $L$ is the solution at $x=0$ and $R$ is the solution at $x=\pi/3$?

If We look at a similar problem, $$d_x^2 g = f(x)\\g(0)=g(1) = 0$$ we see its Green's function is $$G_L = l(x) r(y):0<x<y<\pi/3\\G_R=l(y)r(x):0<y<x<\pi/3$$
where $$l(x) = -x\\ r(x) = (1-x)$$ which makes sense, since $l(0)=0$ and $r(1)=0$.

So why is there a difference (or am I not seeing something) in the two approaches?