The differential equation
\[
d
x
(
t
)
=
a
(
x
(
t
)
,
t
)
d
Z
(
t
)
+
b
(
x
(
t
)
,
t
)
d
t
dx(t) \, = \, a(x(t),t) \,dZ(t) \:+\: b(x(t),t) \,dt
\]
for fractal-type functions
Z
(
t
)
Z(t)
is determined via fractional calculus. Under appropriate conditions we prove existence and uniqueness of a local solution by means of its representation
x
(
t
)
=
h
(
y
(
t
)
+
Z
(
t
)
,
t
)
x(t)\, =\, h(y(t)+Z(t),t)
for certain
C
1
C^1
-functions
h
h
and
y
y
. The method is also applied to Itô stochastic differential equations and leads to a general pathwise representation. Finally we discuss fractal sample path properties of the solutions.