= Solution
Set $\boxed{\xi=s/L,\quad\lambda=PL^2/(\pi^2EI)}$. The endpoint interval becomes $[0,1]$, and multiplication of the dimensional equation by $L^2/EI$ gives
$$
\boxed{\theta_{\xi\xi}+\pi^2\lambda\sin\theta=0,\qquad\theta(0)=\theta(1)=0.}
$$
The energy in these variables is $U=(EI/L)\int_0^1[\theta_\xi^2/2-\pi^2\lambda(1-\cos\theta)]\,d\xi$. Thus $\lambda$ measures load relative to the first linearized <eigenvalue> for the angular endpoint conditions actually imposed.
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