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Does there exist a general solution for a differential equation like: $$\ddot{\phi}(x,t) - \partial^2_x\phi(x,t) + \phi(x,t)m^2(x) = 0,$$ where $m(x)$ is a known function.

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    $\begingroup$ Depends on the form of m(x) $\endgroup$ – 0x90 Oct 6 at 10:56
  • $\begingroup$ It is smooth and has a compact support. I only have a numerical solution which states $m(x)$. $\endgroup$ – AbuuzArbuz Oct 6 at 11:07
  • $\begingroup$ I think that under certain conditions on the function $m(x)$, a solution can be found by using Green's function. But to get a specific solution you should mention the function. $\endgroup$ – Manvendra Somvanshi Oct 6 at 11:38
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The Poschl-Teller equation with $m^2(x)= n(n+1) {\rm sech}^2 x$, $n\in {\mathbb Z}$ is often found in soliton systems. It has closed-form solutions with bound states and scattering states.

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  • $\begingroup$ OP was not asking for special solutions. $\endgroup$ – Hans Moleman Oct 6 at 13:55
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    $\begingroup$ @HansMoleman it’s still a useful answer though. $\endgroup$ – 0x90 Oct 6 at 14:13
  • $\begingroup$ Clearly there can be no closed-form general solution. If there were, there would be a closed-form solution to the general 1-dimensional Schroedinger equation. $\endgroup$ – mike stone Oct 6 at 21:15

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