This leads to the following
definition:
A function F(x, y) is said to be homogeneous function of degree n if
F(λx, λy) = λn F(x, y) for any nonzero constant λ We note that in the above examples, F1, F2, F3 are homogeneous functions of
degree 2, 1, 0 respectively but F4 is not a homogeneous function © NCERT
not to be republished
MATHEMATICS
398
We also observe that
F1(x, y) =
2
2
2
1
2
2
y
y
y
x
x h
x
x
x
⎛
⎞
⎛
⎞
+
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
or
F1(x, y) =
2
2
2
2
1
x
x
y
y h
y
y
⎛
⎞
⎛
⎞
+
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
F2(x, y) =
1
1
3
3
2
y
y
x
x h
x
x
⎛
⎞
⎛
⎞
−
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
or
 F2(x, y) =
1
1
4
2
3
x
x
y
y h
y
y
⎛
⎞
⎛
⎞
−
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
F3(x, y) =
0
0
5
cos y
y
x
x h
x
x
⎛
⎞
⎛
⎞
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
F4(x, y) ≠
6
n
y
x h
x
⎛
⎞
⎜
⎟
⎝
⎠ , for any n ∈ N
or
F4 (x, y) ≠
7
n
x
y h
⎛y
⎞
⎜
⎟
⎝
⎠ , for any n ∈ N
Therefore, a function F (x, y) is a homogeneous function of degree n if
F(x, y) =
or
n
n
y
x
x g
y h
x
y
⎛
⎞
⎛
⎞
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
A differential equation of the form dy
dx = F (x, y)  is said to be homogenous if
F(x, y) is a homogenous function of degree zero To solve a homogeneous differential equation of the type
(
)
F
,
dy
x y
dx =
 =
y
g
x
⎛
⎞
⎜
⎟
⎝
⎠