Laplace equation with mixed conditions

Question

I want to solve the following problem $$u_{xx}+u_{yy}=0,$$ for $0<x<\infty$ and $0<y<\infty$ with the conditions $u(x,0)=0$, $u_x(0,y)+u_y(0,y)=-\frac{q_0}{k}$ for $0<y<L$ and $u_x(0,y)+u_y(0,y)=0$ for $y\ge L$. I end up with a solution on the form $$ u(x,y)=\int_0^{\infty}A(\mu)e^{-\mu x}\sin(\mu y)\,d\mu. \tag{3} $$ with $$u_x(0,y)+u_y(0,y)=\int_0^{\infty}\mu A(\mu)(\cos(\mu y)-\sin(\mu y))\,d\mu =\begin{cases} -\frac{q_0}{k}&\text{if $0<y<L$,} \\ 0&\text{if $y\geq L$.} \end{cases} \tag{4} $$ But i dont know what to do next. Any help will be very appreciated