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Differential equations with infinitely many derivatives, sometimes alsoreferred to as ``nonlocal'' differential equations, appear frequently inbranches of modern physics such as string theory, gravitation and cosmology.The goal of this paper is to show how to properly interpret and solve suchequations, with a special focus on a solution method based on the Boreltransform. This method is a far-reaching generalization of previous approaches(N. Barnaby and N. Kamran, Dynamics with infinitely many derivatives: theinitial value problem. {\em J. High Energy Physics} 2008 no. 02, Paper 008, 40pp.; P. G\'orka, H. Prado and E.G. Reyes, Functional calculus via Laplacetransform and equations with infinitely many derivatives. {\em Journal ofMathematical Physics} 51 (2010), 103512; P. G\'orka, H. Prado and E.G. Reyes,The initial value problem for ordinary equations with infinitely manyderivatives. {\em Classical and Quantum Gravity} 29 (2012), 065017). Inparticular we reconsider generalized initial value problems and disprovevarious conjectures found in the modern literature. We illustrate variousphenomena that can occur with concrete examples, and we also treat efficientimplementations of the theory.

On linear differential equations with infinitely many derivatives