= Solution
Let $s$ be <arc length>, so <inextensibility> makes $r'=d_3$ a <unit tangent vector>. The <curvature> is $\kappa=\theta'$, hence $m'=EI\theta''e_z$ for uniform <bending stiffness>. <Balance of linear momentum> in static equilibrium gives a constant resultant $n=Fe_x+Ge_y$.
Taking the <cross product> directly,
$$
d_3\times n=(\cos\theta\,e_x+\sin\theta\,e_y)\times(Fe_x+Ge_y)
=(G\cos\theta-F\sin\theta)e_z.
$$
The scalar component of <conservation of angular momentum> therefore becomes
$$
\boxed{EI\theta''+G\cos\theta-F\sin\theta=0.}
$$
Together with $x'=\cos\theta$ and $y'=\sin\theta$, this <Euler elastica> equation determines the centerline once the appropriate <boundary conditions> are supplied.
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