Differential equation with phase shift

Question

What are differential equations called where the derivative is not evaluated at the same point as the function?

E.g. $y'(t) = y(t+\pi/2)$ with the solution $y(t) = \sin(t)$.

Answer 1

They are called delay differential equations, except that in your equation $y'$ depends on later values of $y$ rather than earlier values (but you could change $t$ to $-t$ to remedy that).