(4)
Substituting the value of y  and dy
dx in equation (1) we get
dv
v
+x dx
 = 1
12
+vv
−
or
dv
x dx  = 1
12
v
v
+v
−
−
or
dv
x dx  =
2
1
v1
vv
 
or
2
1
1
v
dv
v
v  
 =
dx
x
 
Integrating both sides of equation (5), we get
2
1
1
v
dv
v
 v
 
 
 =
xdx
 
or
2
1
2
1
3
2
1
v
dv
v
 v
 
 
 
 
 = – log |x| + C1
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DIFFERENTIAL EQUATIONS
401
or
1
2
2
1
2
1
3
1
log
C
2
2
1
1
 
  
 
 
 
 
 
 
 
v
dv
dv
x
v
v
v
v
or
2
1
2
2
1
3
1
log
1
log
C
2
2
1
3
2
2
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
v
v
dv
x
v
or
2
1
1
1
3
2
2
1
log
1 tan
log
C
2
2
3
3
 
 
 
 
 
 
  
 
 
 
 
 
v
v
v
x
or
2
2
1
1
1
1
2
1
log
1
log
3 tan
C
2
2
3
 
 
 
 
 
 
 
 
 
 
 
 
v
v
v
x
(Why )
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
⎛
⎞ +
⎜
⎟
⎝
⎛⎠
⎞
⎜
⎟
⎝
⎠