The integrating factor of D.E.
dy/dx+ y. tanx =Cos2x is...
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1) (x
2
+xy)dy–(x
2
+y
2
)dx
dx
dy
=
x
2
+xy
x
2
+y
2
=
1+(
x
y
)
1+(
x
y
)
2
u=
x
y
dx
dy
=
x
1
(
dx
dy
)−
x
2
y
=
x
1
dx
dy
−(
x
u
)
(
dx
du
+
x
u
)x=
dx
dy
(
dx
xdu
+4)=
1+4
1+4
2
dx
xdu
=
1+4
1+4
2
−u=
1+4
1+4
2
−4−4
2
dx
xdu
=(
1+4
1−4
)
⇒(
1−4
1+4
)du=
x
dx
∫
1−4
−4−1
dx=−∫
x
dx
=∫
1−4
−4+1−1−1
du=−∫
x
dx
=∫(1−
1−4
2
)du=−∫
x
dx
u+2ln(1−y)=−ln(x)+c
⇒u+2ln(1−4)=−ln(x)+c
⇒u=
x
y
⇒
x
y
=2ln(1−
x
y
)=−ln(x)+c
⇒y+2xln(1−
x
y
- )=−xln(x)+x
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