Question

$(xy ^{2} - e^{y} {x}^{3} )dx - {x}^{2} ydy = 0$
​

Answer 1

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

Answer 2

Answer:

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