Solving PDEs without Boundary Conditions: A Conundrum?

[zsua]

If a PDE has no boundary conditions specified, how does one go about providing a solution--even if this is a general solution?

I'm stuck looking at the separation of variables and other methods which all seem to heavily rely on those boundary conditions and initial conditions.

If anyone wants to conceptualize what I'm talking about more, it's a two dimensional Poisson equation (inhomogeneous).

 

[HallsofIvy]

In general, where the general solution to an ordinary differential equation involves unknown constants, the general solution to a partial differential equation involves unknown functions.

For example, a function $\phi(x,t)$ satisfies the "wave equation"
$$\frac{\partial^2\phi}{\partial x^2}= \frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}$$
if and only if it is of the form
$$\phi(x, t)= F(x+ ct)+ G(x- ct)$$
where F and G can be any twice differentiable functions.

 

[zsua]

Without boundary conditions, how do you even come up with a solution at all?

Anything specific like separation of variables for one case of the constant?