Given an integral equation is there always a differential equation which has the same (say smooth) solutions? It seems like not but can one prove this in some example?

**Edit:** Naively I'm hoping for some algorithm which takes an integral equation and applies some operations like taking derivatives, substituting variables for some new ones, adding additional differential equations etc... such that after this procedure you have made all integral signs vanish and obtained a differential equation which has the same solutions as the integral eqution. (maybe similarly to how one can transform any system of PDEs into a system of first order equations)