Given task:
Find a solution to the boundary value problem in the domain $\left \{ x>0, y>0 \right \}$ in the class of bounded functions using the Fourier transform $$u_{xx}+4u_{yy}=0, u|_{x=0}=\theta (2-y),u|_{y=0}=0$$
My attempt: I don’t know how to solve it, if the problem is in the boundary conditions, we won’t be able to solve it, there the boundary conditions contradict each other: $u(0;y)=\theta(2-y)$ and $u(x;0)=0$, accordingly $u(0;0)$ must be both $0$ and $1$
Tell me how to solve this problem?