Consider the Laplace equation
$$\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = 0,\quad\quad-\infty<x<\infty,\quad 0<y<\infty$$
with the boundary conditions
$$\lim_{|x|\rightarrow \infty} u(x,y)\rightarrow 0, \quad 0 < y < \infty,$$
$$\lim_{y\rightarrow \infty} u(x,y)\rightarrow 0, \quad -\infty < x < \infty,$$
$$u(x,0)=x, \quad 0 \leq x < 1,$$
$$u_y(x,0)=0, \quad 1 < x < \infty,$$
which is 100% taken from the book of Dean G. Duffy (Transform method for solving Partial differential equations second edition page 264 problem 6). I tried to solve using the separation of variables as suggested by the book but given boundary conditions limit my understanding of finding the general solution. could you please help me out?.