= Solution
Here $L_1=1$ and $L_2=1/2$, so a <perturbation> varying in both coordinates has $n,m\ge1$ and
$$
s=\pi^2(n^2+4m^2)\ge5\pi^2.
$$
With the given unit coefficients the <determinant> becomes $D(s)=(1+s)^2-\beta s$. The <trace> remains negative, so <instability> occurs precisely when $D(s)<0$, or
$$
\beta>s+2+\frac1s.
$$
The right-hand side is strictly increasing for $s>1$, since its <derivative> is $1-s^{-2}>0$. Its smallest allowed value is attained by the mode $(n,m)=(1,1)$, giving the onset threshold
$$
\boxed{\beta_{\rm crit}=5\pi^2+2+\frac1{5\pi^2}\simeq51.3683.}
$$
At this value the first mixed mode has a zero <eigenvalue>; it becomes <unstable> for $\beta>\beta_{\rm crit}$. Thus the critical value is the infimum of the strictly <unstable> range, rather than a strictly <unstable> point itself. Its spatial factor is $\cos(\pi x)\cos(2\pi y)$.
Back to article page
