Seeking Fourier series solution on Laplace equation...still looking, am I on track?

Solution 1:

Yes, everything you did makes sense. It would be easier to work with $e^{\pm nx}\sin ny$ to begin with: these functions solve the equation and satisfy the boundary conditions. The positive exponents are rejected because we want solution to remain bounded (so that it would describe, say, the stable heat distribution in semi-infinite strip). The form of the solution ends up being
$$u(x,y)=\sum_{n=1}^\infty e^{-nx} b_n \sin ny \tag1$$ The coefficients $b_n$ are determined from the Fourier series of the initial data $f$, because plugging $x=0$ into (1) we get $u(0,y)=f(y) = \sum_{n=1}^\infty b_n \sin ny$. If you are expected to find $b_n$, the function $f$ should have been given in the problem.