= Reflection-symmetric trials for a linearly weighted Rayleigh quotient
{title2=$\frac{\int_0^L(y')^2dx}{\int_0^Lxy^2dx}=\frac2L\frac{\int_0^L(y')^2dx}{\int_0^Ly^2dx}$}
For a nonzero <trial function> with $y(0)=y(L)=0$ and $y(L-x)=y(x)$, substituting $L-x$ for $x$ proves $\int_0^Lxy^2dx=(L/2)\int_0^Ly^2dx$. This gives the displayed <Rayleigh quotient>. The <Poincaré inequality> on an interval bounds it below by $2\pi^2/L^3$, with equality for a nonzero multiple of $\sin(\pi x/L)$. Thus this sine is the best reflection-symmetric trial for $-y''=\lambda xy$, while the unrestricted lowest <eigenfunction> need not be reflection-symmetric. The polynomial $x(L-x)$ gives $20/L^3$, only a factor $10/\pi^2$ above the sine's value. Both are upper bounds on the unrestricted lowest <eigenvalue> by the <Rayleigh-Ritz method>.
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