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

Solution for (2x-y+1)dx+(2y-x+1)dy=0 equation:

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

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

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

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

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

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

Reorder the terms:
1dx + -1dxy + 2dx2 + (-1dxy + 1dy + 2dy2) = 0
1dx + -1dxy + 2dx2 + (-1dxy + 1dy + 2dy2) = 0

Reorder the terms:
1dx + -1dxy + -1dxy + 2dx2 + 1dy + 2dy2 = 0

Combine like terms: -1dxy + -1dxy = -2dxy
1dx + -2dxy + 2dx2 + 1dy + 2dy2 = 0

Solving
1dx + -2dxy + 2dx2 + 1dy + 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(x + -2xy + 2x2 + y + 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 '(x + -2xy + 2x2 + y + 2y2)' equal to zero and attempt to solve:

Simplifying
x + -2xy + 2x2 + y + 2y2 = 0

Solving
x + -2xy + 2x2 + y + 2y2 = 0

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

Add '-1x' to each side of the equation.
x + -2xy + 2x2 + y + -1x + 2y2 = 0 + -1x

Reorder the terms:
x + -1x + -2xy + 2x2 + y + 2y2 = 0 + -1x

Combine like terms: x + -1x = 0
0 + -2xy + 2x2 + y + 2y2 = 0 + -1x
-2xy + 2x2 + y + 2y2 = 0 + -1x
Remove the zero:
-2xy + 2x2 + y + 2y2 = -1x

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

Reorder the terms:
-2xy + 2xy + 2x2 + y + 2y2 = -1x + 2xy

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

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

Reorder the terms:
2x2 + -2x2 + y + 2y2 = -1x + 2xy + -2x2

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

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

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

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

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

Simplifying
0 = -1x + 2xy + -2x2 + -1y + -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}