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The standard procedure for computing scalar multi-loop Feynman integralsconsists in reducing them to a basis of so-called master integrals, derivedifferential equations in the external invariants satisfied by the latter and,finally, try to solve them as a Laurent series in $\epsilon = (4-d)/2$, where$d$ are the space-time dimensions. The differential equations are, in general,coupled and can be solved using Euler's variation of constants, provided that aset of homogeneous solutions is known. Given an arbitrary differential equationof order higher than one, there exist no general method for finding itshomogeneous solutions. In this paper we show that the maximal cut of theintegrals under consideration provides one set of homogeneous solutions,simplifying substantially the solution of the differential equations.

On the maximal cut of Feynman integrals and the solution of their differential equations