)
Replacing v by y
x , we get
or
2
2
1
1
2
1
1
2
log
1
log
3 tan
C
2
2
3
 
 
 
 
 
 
 
 
 
 
 
 
 
y
y
y
x
x
x
x
x
or
2
2
1
1
2
1
2
log
1
3 tan
C
2
3
y
y
y
x
x
x
x
x
−
⎛
⎞
+
⎛
⎞
+
+
=
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
or
2
2
1
1
2
log (
)
2 3 tan
2C
y3
x
y
xy
x
x
−
+
⎛
⎞
+
+
=
+
⎜
⎟
⎝
⎠
or
2
2
1
2
log (
)
2 3 tan
C
3
−
+
⎛
⎞
+
+
=
+
⎜
⎟
⎝
⎠
x
y
x
xy
y
x
which is the general solution of the differential equation (1)
Example 16 Show that the differential equation 
cos
cos
y
dy
y
x
y
x
x
dx
x
⎛
⎞
⎛
⎞
=
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
 is
homogeneous and solve it Solution The given differential equation can be written as
dy
dx  =
cos
cos
y
y
x
x
y
x
x
⎛
⎞ +
⎜
⎟
⎝
⎛⎠
⎞
⎜
⎟
⎝
⎠ (1)
© NCERT
not to be republished
MATHEMATICS
402
It is a differential equation of the form 
F( ,
)
dy
x y
dx = Here
F(x, y) =
cos
cos
y
y
x
x
y
x
x
⎛
⎞ +
⎜
⎟
⎝
⎛⎠
⎞
⎜
⎟
⎝
⎠
Replacing x by λx and y by λy, we get
F(λx, λy) =
0
[ cos
]
[F( , )]
cos
y
y
x
x
x y
y
x
x
⎛
⎞
λ
+
⎜
⎟
⎝
⎠
= λ
⎛
⎞
λ⎜
⎟
⎝
⎠
Thus, F(x, y) is a homogeneous function of degree zero