I am trying to find the general solution for the following special case of the Klein-Gordon equation:
$$ {\partial^2f(x,y)\over{\partial x^2}} -c^2{\partial^2f(x,y)\over{\partial y^2}}+g(x)f(x,y)=0\tag1$$
where $x$ and $y$ are the general coordinates, $c$ is a constant, $g(x)$ is a known function of only $x$ and $f(x,y)$ is the unknown function which represents the general solution of equation $(1)$.
Here, I found some particular solutions to equation $(1)$ for the case where $g(x)$ is not a function of $x$, but a constant. Also, here can be found a general solution to the Klein-Gordon equation, but also for the case where $g(x) = const$.
Since I did not find in literature the general solution to equation $(1)$, I tryed deriving it myself. However, I did not cucceed. So my questions are:
- Does a general solution to equation $(1)$ even exist? If it does, what is it?
I need to answer these questions so I can design the funtion $g(x)$ for my project. Thank you for your time.
Let's follow Helmholtz' classic method which leads from the wave equation to the Helmholtz equation:
Separating variables $f(x,y)=u(x)v(y)$ we get \begin{align} u''v-c^2uv''+guv=0\,, \end{align} that is, \begin{align} \frac{u''}{u}+g=c^2\frac{v''}{v}\,. \end{align} The left hand side depends only on $x$ and the right hand side only on $y$. Therefore there must be a constant $\lambda$ such that \begin{align}\tag{1} \frac{u''}{u}+g=\lambda\,,\quad c^2\frac{v''}{v}=\lambda. \end{align} Can you solve these two linear ODEs ?
The general solution for your linear Klein-Gordon equation will be a superposition of all solutions of (1), i.e., an integral over $\lambda$.