
1186 CAUCHY PROBLEM. FUNDAMENTAL SOLUTIONS
◮ Classical and generalized C auchy problems for ordinary differential equations.
Consider the Cauchy problem for the constant coefficient linear equation (16.2.1.1) (with
a
k
= const) with the initial conditions
w
(k)
t
= b
k
at t = 0; k = 0, 1, . . . , m − 1, (16.2.1.5)
where the b
k
are some constants.
Let w(t) be the classical solution of the Cauchy problem for t > 0. We extend the
functions w(t) and f(t) by zero into the domain t < 0. Let us denote the extended functions
by w
+
and f
+
. We have
w
+
= ϑ(t)w(t), f
+
= ϑ(t)f (t). (16.2.1.6)
By successively differentiating the first relation in (16.2.1.6) and by taking into