Solution The given differential equation can be written as
2
2
sin
cos
cos
sin
y
y
y
y
x y
x
dy
xy
y
dx
x
x
x
x
⎡
⎤
⎡
⎤
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
−
=
+
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎢
⎥
⎢
⎥
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
⎣
⎦
⎣
⎦
or
dy
dx  =
2
2
cos
sin
sin
cos
y
y
xy
y
x
x
y
y
xy
x
x
x
⎛
⎞
⎛
⎞
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎛
⎞
⎛
⎞
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
Dividing numerator and denominator on RHS by x2, we get
dy
dx  =
2
2
cos
sin
sin
cos
y
y
y
y
x
x
x
x
y
y
y
x
x
x
⎛
⎞
⎛
⎞
⎛
⎞
+
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
⎛
⎞
⎛
⎞
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠ (1)
Clearly, equation (1) is a homogeneous differential equation of the form dy
y
g
dx
⎛x
⎞
=
⎜
⎟
⎝
⎠ To solve it, we make the substitution
y = vx (2)
or
dy
dx  =
dv
v
x dx
+
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MATHEMATICS
418
or
dv
v
+x dx
 =
2
cos
sin
sin
cos
v
v
v
v
v
v
v
+
−
(using (1) and (2))
or
dv
x dx  =
sin2 cos
cos
v
v
v
v
v
−
or
sin
coscos
v
v
v
dv
v
−v
⎛
⎞
⎜
⎟
⎝
⎠
 = 2 dx
x
Therefore
sin
coscos
v
v
v dv
v
−v
⎛
⎞
⎜
⎟
⎝
⎠
∫
 =
1
2
xdx
∫
or
1
tanv dv
vdv
−
∫
∫
 =
1
2
xdx
∫
or
log sec
log | |
v
v
−
 =
1
2log |
|
log | C |
x +
or
logsec2
v xv
 = log |C1|
or
secv2
v x  = ± C1