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

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


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

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

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

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

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

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

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

Solving
-4dxy + dx2 + dy2 = 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 + y2) = 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 + y2)' equal to zero and attempt to solve:

Simplifying
-4xy + x2 + y2 = 0

Solving
-4xy + x2 + y2 = 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 + y2 = 0 + 4xy

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

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

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

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

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

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

Simplifying
0 = 4xy + -1x2 + -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}

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