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I'm trying to find all solutions of:

[tex]\frac{\partial u}{\partial t}(x,t)-\frac{\partial u}{\partial x}(x,t)=-2u(x,t)[/tex]

I know that one solution is [tex]u(x,t)=Ae^{x-t}[/tex], and any solution of [tex]\frac{\partial u}{\partial t}(x,t)-\frac{\partial u}{\partial x}(x,t)=0[/tex] is of the form [tex]u(x,t)=f(x+t)[/tex].

I tried adding these solutions together but it doesn't work...

The question says that a change of coordinates to simplify [tex]\frac{\partial u}{\partial t}(x,t)-\frac{\partial u}{\partial x}(x,t)[/tex] could be useful, but I don't know where to begin in doing that...

Any help would be much appreciated, thanks.

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# Solution of a Transport-type equation

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