Question

Soive (x2 - 4xy – 2y)dx + (y2 - 4xy – 2x2)dy = 0​

Answer 1

Answer:

Let, M=x

2

−4xy−2y

2

N=y

2

−4xy−2x

2

∴

∂y

∂M

=−4x−4y

and

∂x

∂N

=−4y−4x

Here,

∂y

∂M

=

∂x

∂N

So, the equation is exact.

∴∫Mdx=∫(x

2

−4xy−2y

2

)dx

=

3

x

3

−

2

4x

2

y−2y

2

x

=

3

x

3

−2x

2

y−2y

2

x

And, N free of x=y

2

∴∫y

2

dy

=

3

y

3

∴ the solution is

3

x

3

−2x

2

y−2y

2

x+

3

y

3

=c.