dx
x
x y
h
dy
⎛y
⎞
=
=
⎜
⎟
⎝
⎠
Example 15  Show that the differential equation (x – y) dy
dx  = x + 2y is homogeneous
and solve it Solution The given differential equation can be expressed as
dy
dx  =
2
x
y
x
+y
− (1)
Let
F(x, y) =
2
x
y
x
 y
 
Now
F(λx, λy) =
0
(
2 )
( , )
(
)
x
y
F x y
x
y
 
  
 
 
 
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not to be republished
MATHEMATICS
400
Therefore, F(x, y) is a homogenous function of degree zero So, the given differential
equation is a homogenous differential equation