Example 17 Show that the differential equation 2
2
0
x
x
y
y
y e dx
y
x e
dy
⎛
⎞
⎜
⎟
+
−
=
⎝
⎠
is
homogeneous and find its particular solution, given that, x = 0 when y = 1 Solution The given differential equation can be written as
dx
dy  = 2
2
x
y
x
y
x e
y
y e
− (1)
Let
F(x, y) = 2
2
x
y
x
y
xe
y
ye
−
Then
F(λx, λy) =
0
2
[F( , )]
2
x
y
x
y
xe
y
x y
ye
⎛
⎞
⎜
⎟
λ
−
⎜
⎟
⎝
⎠ =λ
⎛
⎞
⎜
⎟
λ⎜
⎟
⎝
⎠
Thus, F(x, y) is a homogeneous function of degree zero Therefore, the given
differential equation is a homogeneous differential equation