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I remember that when I was studying differential equations, there was an example with solutions of the form $f(x) + C_1$ for $x>0$ and $f(x)+C_2$ for $x<0$ where $C_1$ and $C_2$ may be different for the same solution.

First, could you remind a simple differential equations with such a solution with different constants form positive and negative $x$?

Second, can a similar thing hold for PDE? (I mean non-trivial PDE, that is not obviously equivalent to an ordinary differential equations.) Could you give examples?

Third, are there examples of such PDE (with different constants for different sub-domains) in physics (especially in quantum stuff, or maybe in general relativity)? I am especially interested in PDE expressing a hypothetical structure for which there are no (non-experimental) argument whether constants may be different, that is the proposed theory would be different dependently on whether we allow different constants for different parts of a solution.

Well, I am also interested in the cases when in physics it is required that constants for the same solution are the same, when this does not follow from a PDE itself but follows from some physical considerations.

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