= Solution
Expand the constant right-edge value in a <Fourier sine series>. Its coefficients are
$$
b_n=\frac2\pi\int_0^\pi\sin(ny)dy=\frac{2[1-(-1)^n]}{n\pi}.
$$
The <superposition principle> combines the normalized modes, giving
$$
\boxed{f(x,y)=\frac4\pi\sum_{m=0}^\infty\frac{\sinh[(2m+1)x]}{(2m+1)\sinh[(2m+1)\pi]}\sin[(2m+1)y].}
$$
This is a <harmonic function in a rectangle with one constant boundary value>, with the coordinate roles interchanged. The series and its derivatives converge uniformly on compact subsets of $0<x<\pi$, so it satisfies <Laplace's equation> there. It vanishes on the other three edges and approaches one at every interior point of the right edge. At the two right corners the one-sided prescribed values disagree; no continuous extension can satisfy both. The stated open-edge condition avoids that incompatibility, and isolated corner assignments do not change the bounded interior <Dirichlet problem>.
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