A modified form of the Alekseev variation of constants equation is used to relate the solutions of systems of the form
x
˙
=
f
(
t
,
x
,
λ
)
,
λ
\dot x = f(t,x,\lambda ),\lambda
in
R
m
{R^m}
and the perturbed system
y
˙
=
f
(
t
,
y
,
ψ
(
t
)
)
+
g
(
t
,
y
)
\dot y = f(t,y,\psi (t)) + g(t,y)
. Hypotheses are given on the
m
m
parameter family of differential equations
x
˙
=
f
(
t
,
x
,
λ
)
\dot x = f(t,x,\lambda )
so that if
ψ
˙
\dot \psi
and
g
g
are perturbation functions, bounds can be calculated for the solutions of the perturbed system.