= Solution
Take the clamp at $s=0$ with $x(0)=y(0)=\theta(0)=0$. An end <force> without an applied end moment gives the free-end condition $\theta'(1)=0$. With $F=0$, the <Euler elastica> equation is $EI\theta''+G\cos\theta=0$. For $|G|/(EI)\ll1$, $\theta$ is small, so the leading equation is
$$
EI\theta''=-G.
$$
Integrating with the two <boundary conditions> gives
$$
\boxed{\theta(s)=\frac G{EI}\left(s-\frac{s^2}{2}\right).}
$$
To first order, $x'=\cos\theta=1+O(G^2)$ and $y'=\sin\theta=\theta+O(G^3)$. Hence
$$
\boxed{x(s)=s+O(G^2),\qquad
y(s)=\frac{G}{6EI}s^2(3-s)+O(G^3),\quad0\leq s\leq1.}
$$
Equivalently $y(x)=Gx^2(3-x)/(6EI)$ at leading order, the familiar <cantilever> response. Because $G<0$, the deflection is downward. The free-end <displacement field> and slope are $y(1)=G/(3EI)$ and $\theta(1)=G/(2EI)$. The remainder notation holds with fixed positive $EI$ and nondimensional unit rod length.
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