= Solution
The <superposition principle> gives the <general solution> as $y_c+2y_{p,2}+y_{p,1}$:
$$
y=e^{-\sqrt3x/2}\left(A\cos\frac{3x}{2}+B\sin\frac{3x}{2}\right)+\frac{2x}{3}-\frac2{3\sqrt3}+\frac13e^{-\sqrt3x}.
$$
The first <boundary condition> gives $A=2/(3\sqrt3)-1/3$. At $x=\pi$, the <cosine> vanishes and the <sine> equals $-1$. Comparing with the second <boundary condition> then gives
$$
-Be^{-\sqrt3\pi/2}+\frac{2\pi}{3}=0,
\qquad B=\frac{2\pi}{3}e^{\sqrt3\pi/2}.
$$
The unique solution satisfying both <boundary conditions> is therefore
$$
\boxed{y=e^{-\sqrt3x/2}\left[\left(\frac2{3\sqrt3}-\frac13\right)\cos\frac{3x}{2}+\frac{2\pi}{3}e^{\sqrt3\pi/2}\sin\frac{3x}{2}\right]+\frac{2x}{3}-\frac2{3\sqrt3}+\frac13e^{-\sqrt3x}.}
$$
Uniqueness follows because a <homogeneous solution> zero at both endpoints would have $A=0$ from the first and $B=0$ from the second.
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