I'm trying to solve the PDE problem
$$u_{xx} + u_{yy} -u = 0$$
where $$0<x<\infty, \space 0<y<\infty, \space u(0,y) = 0, \space\lim_{(x^2+y^2)\to \infty}u(x,y) = 0, \space u(x,0) = \begin{cases} 1 & \text{for $0<x\le 1$}\\ 0 & \text{for $x\ge1$}\end{cases}$$ using the Fourier sine transform. Here's what I've done so far: $$\begin{align} \mathcal{F}_s\left(\frac{\partial^2u}{\partial x^2}\right) &= \int_0^\infty\frac{\partial^2 u}{\partial x^2} \sin(kx) dx\\ &= \left[\frac{\partial u}{\partial x}\sin(kx)\right]^\infty_0-k\int_0^\infty \frac{\partial u}{\partial x}\cos(kx)dx\\ &= -k\mathcal{F}_c\left(\frac{\partial u}{\partial x}\right)\\ &= -k(k\mathcal{F}_s(u(x,y))-u(0,y))\\ &= -k^2 \hat u(k,y) - u(0,y)\\ &= -k^2 \hat u(k,y)\\ \end{align}$$ $$\begin{align} \mathcal{F}_s\left(\frac{\partial^2 u}{\partial y^2}\right) &= \int_0^\infty\frac{\partial^2 u}{\partial x^2} \sin(kx)dx\\ &= \frac{\partial^2}{\partial y^2}\int_0^\infty u(x,y) \sin(kx)dx\\ &=\frac{\partial^2}{\partial y^2}\hat u(k,y) \end{align}$$
$$\mathcal{F}_s(u(x,y)) = \hat u(k,y)$$
$$\therefore \frac{\partial^2}{\partial y^2}\hat u(k,y) = \hat u(k,y)(1+k^2)$$
This looks like an ordinary differential equation, so I solve by finding the auxiliary equation with undetermined functions of $k$:
$$\begin{align} \lambda ^2 - (1+k^2) &= 0 \implies \lambda = \pm\sqrt{1+k^2}\\ \therefore \hat u(k,y) &= A(k) \exp({y \sqrt{1+k^2}}) + B(k) \exp({-y \sqrt{1+k^2}})\\ \end{align} $$
Then all is left is to find the functions $A(k)$ and $B(k)$, and take the inverse Fourier transform. Here's where I get stuck. I attempt to take the transform of the piecewise boundary condition (not sure if this part is correct):
$$\begin{align} \hat u(x,0) &= \begin{cases} \mathcal{F}_s(1) & \text{for $0<x\le 1$}\\ 0 & \text{for $x\ge1$}\end{cases}\\ \mathcal{F}_s (1) &= \int_0^1 \sin(kx)dx\\ &= \left[-\frac1k \cos(kx)\right]^1_0\\ &= \frac1k(1-\cos k) \end{align} $$ Now I'm not sure how to proceed. The other boundary condition doesn't seem to be useful in calculating the two functions, and I don't know what to do next. Am I on the right lines? If so, how can I calculate $A(k)$ and $B(k)$?