F = 
cot
log sin
sin
x dx
x
e
e
x
 
 
 
Hence, the solution of the differential equation is given by
y sin x = ∫(2x + x2 cot x) sin x dx + C
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MATHEMATICS
412
or
y sin x = ∫ 2x sin x dx + ∫x2 cos x dx + C
or
y sin x =
2
2
2
2
2
sin
cos
cos
C
2
2
x
x
x
x
dx
x
x dx
⎛
⎞
⎛
⎞
−
+
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∫
∫
or
y sin x =
2
2
2
sin
cos
cos
C
x
x
x
x dx
x
x dx
−
+
+
∫
∫
or
y sin x = x2 sin x + C (1)
Substituting y = 0 and 
2
x
=π
 in equation (1), we get
0 =
2
sin
C
2
2
π
π
⎛
⎞
⎛
⎞ +
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
or
C =
2
4
−π
Substituting the value of C in equation (1), we get
y sin x =
2
2 sin
4
x
x
π
−
or
y =
2
2
(sin
0)
4 sin
x
x
πx
−
≠
which is the particular solution of the given differential equation Example 23 Find the equation of a curve passing through the point (0, 1)