3x(xy-2)dx+((x^3)+2y)dy=0

Solution for 3x(xy-2)dx+((x^3)+2y)dy=0 equation:

Simplifying
3x(xy + -2) * dx + ((x3) + 2y) * dy = 0

Reorder the terms:
3x(-2 + xy) * dx + ((x3) + 2y) * dy = 0

Reorder the terms for easier multiplication:
3x * dx(-2 + xy) + ((x3) + 2y) * dy = 0

Multiply x * dx
3dx2(-2 + xy) + ((x3) + 2y) * dy = 0
(-2 * 3dx2 + xy * 3dx2) + ((x3) + 2y) * dy = 0
(-6dx2 + 3dx3y) + ((x3) + 2y) * dy = 0
-6dx2 + 3dx3y + (x3 + 2y) * dy = 0

Reorder the terms for easier multiplication:
-6dx2 + 3dx3y + dy(x3 + 2y) = 0
-6dx2 + 3dx3y + (x3 * dy + 2y * dy) = 0
-6dx2 + 3dx3y + (dx3y + 2dy2) = 0

Combine like terms: 3dx3y + dx3y = 4dx3y
-6dx2 + 4dx3y + 2dy2 = 0

Solving
-6dx2 + 4dx3y + 2dy2 = 0

Solving for variable 'd'.

Move all terms containing d to the left, all other terms to the right.

Factor out the Greatest Common Factor (GCF), '2d'.
2d(-3x2 + 2x3y + y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor 'd' equal to zero and attempt to solve:

Simplifying
d = 0

Solving
d = 0

Move all terms containing d to the left, all other terms to the right.

Simplifying
d = 0
Subproblem 2
Set the factor '(-3x2 + 2x3y + y2)' equal to zero and attempt to solve:

Simplifying
-3x2 + 2x3y + y2 = 0

Solving
-3x2 + 2x3y + y2 = 0

Move all terms containing d to the left, all other terms to the right.

Add '3x2' to each side of the equation.
-3x2 + 2x3y + 3x2 + y2 = 0 + 3x2

Reorder the terms:
-3x2 + 3x2 + 2x3y + y2 = 0 + 3x2

Combine like terms: -3x2 + 3x2 = 0
0 + 2x3y + y2 = 0 + 3x2
2x3y + y2 = 0 + 3x2
Remove the zero:
2x3y + y2 = 3x2

Add '-2x3y' to each side of the equation.
2x3y + -2x3y + y2 = 3x2 + -2x3y

Combine like terms: 2x3y + -2x3y = 0
0 + y2 = 3x2 + -2x3y
y2 = 3x2 + -2x3y

Add '-1y2' to each side of the equation.
y2 + -1y2 = 3x2 + -2x3y + -1y2

Combine like terms: y2 + -1y2 = 0
0 = 3x2 + -2x3y + -1y2

Simplifying
0 = 3x2 + -2x3y + -1y2

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
d = {0}