(x-2y)dx+2(y-x)dy=0

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

Simplifying
(x + -2y) * dx + 2(y + -1x) * dy = 0

Reorder the terms for easier multiplication:
dx(x + -2y) + 2(y + -1x) * dy = 0
(x * dx + -2y * dx) + 2(y + -1x) * dy = 0

Reorder the terms:
(-2dxy + dx2) + 2(y + -1x) * dy = 0
(-2dxy + dx2) + 2(y + -1x) * dy = 0

Reorder the terms:
-2dxy + dx2 + 2(-1x + y) * dy = 0

Reorder the terms for easier multiplication:
-2dxy + dx2 + 2dy(-1x + y) = 0
-2dxy + dx2 + (-1x * 2dy + y * 2dy) = 0
-2dxy + dx2 + (-2dxy + 2dy2) = 0

Reorder the terms:
-2dxy + -2dxy + dx2 + 2dy2 = 0

Combine like terms: -2dxy + -2dxy = -4dxy
-4dxy + dx2 + 2dy2 = 0

Solving
-4dxy + dx2 + 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), 'd'.
d(-4xy + x2 + 2y2) = 0

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 '(-4xy + x2 + 2y2)' equal to zero and attempt to solve:

Simplifying
-4xy + x2 + 2y2 = 0

Solving
-4xy + x2 + 2y2 = 0

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

Add '4xy' to each side of the equation.
-4xy + x2 + 4xy + 2y2 = 0 + 4xy

Reorder the terms:
-4xy + 4xy + x2 + 2y2 = 0 + 4xy

Combine like terms: -4xy + 4xy = 0
0 + x2 + 2y2 = 0 + 4xy
x2 + 2y2 = 0 + 4xy
Remove the zero:
x2 + 2y2 = 4xy

Add '-1x2' to each side of the equation.
x2 + -1x2 + 2y2 = 4xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + 2y2 = 4xy + -1x2
2y2 = 4xy + -1x2

Add '-2y2' to each side of the equation.
2y2 + -2y2 = 4xy + -1x2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
0 = 4xy + -1x2 + -2y2

Simplifying
0 = 4xy + -1x2 + -2y2

The solution to this equation could not be determined.

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