I'm interested in equations like $y'(x) = y(x+1)$. Note that the argument of the function $y$ on the right hand side of that last equation is $x+1$. The right hand side does not contain a multiplication. We have a function $y$ that takes an argument $x$ which is written as $y(x)$. So the derivative of the solution is equal to the value of the solution "a little bit to the right". I do have an application for this, in case somebody would ask, but I'm interested most of all in the general theory instead of the specific instance (so no XY-problem; thanks Xander H.!).
I can't find the correct name for this kind of equations. Can somebody provide me a pointer?