Given an [integral equation](http://en.wikipedia.org/wiki/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)



Can an integral equation always be rewritten as a differential equation?
While I second Deane's comment that the author should be a bit more specific about the kind of equations he is interested in, in general the answer is **no** for integral and, more broadly,*integro-differential* equations. However, the latter can be reduced to *functional-differential* equations rather than to purely differential ones. For more details, see Section 6.6 of the book *[Symmetries and Conservation Laws for Differential Equations of Mathematical Physics](http://books.google.com/books?id=lXu1s6sxjYUC)*
(unfortunately the relevant pages appear to be excluded from Google preview).


For instance, I greatly doubt that one could reduce the [Smoluchowski coagulation equation](http://en.wikipedia.org/wiki/Smoluchowski_coagulation_equation) from Example 6.5 of the above book to a differential (as opposed to functional-differential) equation or system thereof. 


