Question

If ma3 and n=2, find
the value of menm​

Answer 1

Answer:

(x

2

y−2xy

2

)dx=(x

3

−3x

2

y)dy

(x

2

y−2xy

2

)dx−(x

3

−3x

2

y)dy=0

Here,

M=x

2

y−2xy

2

⇒

∂y

∂M

=x

2

−4xy

N=−(x

3

−3x

2

y)⇒

∂x

∂N

=6xy−3x

2

∵

∂y

∂M



=

∂x

∂N

Therefore,

Integrating factor =

Mx+Ny

1

=

x(x

2

y−2xy

2

)+y(3x

2

y−x

3

)

1

=

x

2

y

2

1

Multiplying the given equation by the integrating factor, we get

(

x

2

y

2

1

)(x

2

y−2xy

2

)dx−(

x

2

y

2

1

)(x

3

−3x

2

y)dy=0

⇒(

y

1

−

x

2

)dx−(

y

2

x

−

y

3

)dy=0

Now again,

M=(

y

1

−

x

2

)⇒

∂y

∂M

=−

y

2

1

N=−(

y

2

x

−

y

3

)⇒

∂x

∂N

=−

y

2

1

Now the above equation is an exact differential equation.

Therefore,

Solution of the equation is-

∫Mdx+∫(terms in N not containing x)dy=C

⇒∫(

y

1

−

x

2

)dx+∫(

y

3

)dy=C

⇒

y

x

−2logx+3logy=C

⇒

y

x

−logx

2

+logy

3

=C

⇒

y

x

+log(

x

2

y

3

)=C