  We apply a recently suggested new strategy to solve differential equations
for Feynman integrals. We develop this method further by analyzing asymptotic
expansions of the integrals. We argue that this allows the systematic
application of the differential equations to single-scale Feynman integrals.
Moreover, the information about singular limits significantly simplifies
finding boundary constants for the differential equations. To illustrate these
points we consider two families of three-loop integrals. The first are
form-factor integrals with two external legs on the light cone. We introduce
one more scale by taking one more leg off-shell, $p_2^2\neq 0$. We analytically
solve the differential equations for the master integrals in a Laurent
expansion in dimensional regularization with $\epsilon=(4-D)/2$. Then we show
how to obtain analytic results for the corresponding one-scale integrals in an
algebraic way. An essential ingredient of our method is to match solutions of
the 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 to
expansion by regions. The second family consists of four-point non-planar
integrals with all four legs on the light cone. We evaluate, by differential
equations, all the master integrals for the so-called $K_4$ graph consisting of
four external vertices which are connected with each other by six lines. We
show how the boundary constants can be fixed with the help of the knowledge of
the singular limits. We present results in terms of harmonic polylogarithms for
the corresponding seven master integrals with six propagators in a Laurent
expansion in $\epsilon$ up to weight six.
