Solving a PDE with boundary problem

[Karl86]

Homework Statement

I want to find the solution to the following problem:
$$\begin{cases} \nabla^2 B=c^2 B &\text{ on the half plane } x>0 \\ B=B_0 \hat{z} & \text{ for } x<0 \end{cases}$$
in the $xz$ plane. $c, B_0 \in \mathbb{R}$

Homework Equations

I am not really sure what would be relevant. I could solve this if I knew that B is a function of only one variable
but it can a priori be a function of $x,y,z$.

The Attempt at a Solution

I know the solution to be of the form $C_1 e^{\frac{x}{d}} + C_2 e^{-\frac{x}{d}}$. But I have no idea how to prove it. I have not really taken a proper course in PDEs. In particular it's not clear to me why the solution has to depend only on x, for example.

 

[Orodruin]

In particular it's not clear to me why the solution has to depend only on x, for example.

What happens to the problem if you make the translation $y \to y + y_0$?

 

[Karl86]

What happens to the problem if you make the translation $y \to y + y_0$?

Good point. The $z$ translation is a bit more problematic to me.

 

[Orodruin]

Good point. The zzz translation is a bit more problematic to me.

Why? The argument is 100 % the same.

 

[Karl86]

Why? The argument is 100 % the same.

Oops. Thanks.