= Solution
After multiplying the initial field by $e^{\beta x/2}$, the $y$ dependence is exactly the $n=2$ cosine mode. Its <orthogonality> therefore gives $A_{mn}=0$ when $n\neq2$. For $n=2$, the remaining <Fourier coefficient> in the <Fourier sine series> is
$$
A_{m2}=2\int_0^1x(1-x)\sin(m\pi x)\,dx=\frac{4[1-(-1)^m]}{m^3\pi^3}.
$$
Thus
$$
\boxed{A_{mn}=\begin{cases}8/(m^3\pi^3),&m\text{ odd and }n=2,\\0,&\text{otherwise}.\end{cases}}
$$
The coefficients are independent of $\beta$, because the initial exponential exactly cancels the drift-removing weight. Substitution into the preceding expansion gives the full subsequent solution, with decay rates $m^2\pi^2+9\pi^2/4+\beta^2/4$.
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