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We apply a recently suggested new strategy to solve differential equationsfor Feynman integrals. We develop this method further by analyzing asymptoticexpansions of the integrals. We argue that this allows the systematicapplication of the differential equations to single-scale Feynman integrals.Moreover, the information about singular limits significantly simplifiesfinding boundary constants for the differential equations. To illustrate thesepoints we consider two families of three-loop integrals. The first areform-factor integrals with two external legs on the light cone. We introduceone more scale by taking one more leg off-shell, $p_2^2\neq 0$. We analyticallysolve the differential equations for the master integrals in a Laurentexpansion in dimensional regularization with $\epsilon=(4-D)/2$. Then we showhow to obtain analytic results for the corresponding one-scale integrals in analgebraic way. An essential ingredient of our method is to match solutions ofthe differential equations in the limit of small $p_2^2$ to our results at$p_2^2\neq 0$ and to identify various terms in these solutions according toexpansion by regions. The second family consists of four-point non-planarintegrals with all four legs on the light cone. We evaluate, by differentialequations, all the master integrals for the so-called $K_4$ graph consisting offour external vertices which are connected with each other by six lines. Weshow how the boundary constants can be fixed with the help of the knowledge ofthe singular limits. We present results in terms of harmonic polylogarithms forthe corresponding seven master integrals with six propagators in a Laurentexpansion in $\epsilon$ up to weight six.

Evaluating single-scale and/or non-planar diagrams by differential equations