I have a PDE in the domain $z\in (0,\infty) $:
$$\frac{\partial^2 W(x,z)}{\partial x^2} + \frac{\partial^2 W(x,z)}{\partial z^2} + \frac{1}{l^2}W(x,z)=0$$
Now, through the variable separable method I introduce $W(x,z)=Q(x)G(z)$ and substitute in the above equation. Thereby,
$$G(z)\frac{\partial^2 Q}{\partial x^2} + Q(x)\frac{\partial^2 G}{\partial z^2} + \frac{1}{l^2}Q(x)G(z)=0$$
Diving by $Q(x)G(z)$ on both sides and rearranging the terms we get
$$\frac{l^2}{Q(x)}\frac{\partial^2 Q}{\partial x^2} + \frac{l^2}{G(z)}\frac{\partial^2 G}{\partial z^2} + 1=0$$
Now, I introduce the separation constant $\lambda$ and then I get the two ODE's in the following form:
$$\frac{\partial^2 Q}{\partial x^2} +\frac{\lambda Q(x)}{l^2} = 0$$ $$\frac{\partial^2 G}{\partial z^2} +\frac{\lambda G(z)}{l^2} + 1 = 0$$
But, the solution is in the form of $\exp(ik_xx + ik_yx + wt)$, where $r$ is the positve solution of a quadratic equation. Can someone suggest how to proceed further? and Where I have done wrong? Thanks for the attention. Here, the initial condition is $W=0$ for $x=z=0$.