(xy^2-xe^1/x^2)dx-x^2ydy=0
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Answer:
Given,
(xy
2
−e
x
3
1
)dx=x
2
ydy
x
2
y
dx
dy
−xy
2
=−e
x
3
1
−(1)
let y
2
=t
∴2ydy=dt
(1)→
2dx
x
2
dt
−xt=−e
−
x
3
1
dx
dt
−
x
2t
=−
x
2
2
e
−
x
3
1
P=
x
−2
,Q=−
x
3
2
e
−
x
3
1
I.f.=e
∫p.dx
=e
∫
x
−2
dx
=e
−2logx
=elogx
−2
=x
−2
Com-plete solution
t×x
−2
=∫x
−2
×−
x
2
2
e
x
3
−1
dx+c
t×x
−2
=−∫
x
4
2
e
x
3
−1
dx+c
Let
x
3
−1
=u
→
x
4
3
dx=du
t×x
−2
=−∫
3
2
e
u
du+c
t×x
−2
=−
3
2
e
u
+c
Substitute the value of t and u
∴y
2
×x
−2
=−
3
2
e
x
3
−1
+c
2x
2
y
2
+
3
1
e
x
3
−1
=c
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