The nonlinear second order differential equation satisfied by the homogeneous function
y
=
[
a
u
m
+
m
b
u
j
v
n
+
c
v
m
]
k
/
m
,
m
=
j
+
n
y = {[a{u^m} + mb{u^j}{v^n} + c{v^m}]^{k/m}},m = j + n
, is obtained. Functions
u
u
and
v
v
satisfy independently the linear equation
y
¨
+
r
(
t
)
y
˙
+
q
(
t
)
y
=
0
\ddot y + r(t)\dot y + q(t)y = 0
. The nonlinear equation derived contains previous results as special cases of
r
(
t
)
r(t)
, of the constants
a
,
b
a,b
, and
c
c
, and of the numbers
k
k
and
m
m
.