Find the general solution of the linear equation: y dx + -2(x+y^6) dy = 0?

Let M = y , N = -2( x + y^6 )
∂M/∂y - ∂N/∂x = 1 - (-2) = 3
( ∂M/∂y - ∂N/∂x ) / M = 3/y ≡ g(y)

Integrating Factor
= I (y)
= e^[ - ∫ g(y) dy ]
= e^[ - ∫ (3/y) dy ]
= e^( - 3*ln y )
= y^(-3)

Hence,
y*y^(-3) dx - 2( x + y^6 )*y^(-3) dy = 0 is an exact O.D.E
That is,
y^(-2) dx - 2[ xy^(-3) + y^3 ] dy = 0 is exact

Φ = ∫ y^(-2) dx = xy^(-2) + f(y)
∂Φ/∂y = - 2xy^(-3) + f ' (y) = - 2[ xy^(-3) + y^3 ]
f ' (y) = - 2y^3
f(y) = - (1/2)y^4 + c
Φ = xy^(-2) + f(y) = xy^(-2) - (1/2)y^4 + c

Ans: xy^(-2) - (1/2)y^4 = c