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

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

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

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

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

Reorder the terms:
(dxy + 1dy + -2dy2) = (2x + -1y + 1) * dx
(dxy + 1dy + -2dy2) = (2x + -1y + 1) * dx

Reorder the terms:
dxy + 1dy + -2dy2 = (1 + 2x + -1y) * dx

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

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

Solving
dxy + 1dy + -2dy2 = 1dx + -1dxy + 2dx2

Solving for variable 'd'.

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

Add '-1dx' to each side of the equation.
dxy + 1dy + -1dx + -2dy2 = 1dx + -1dxy + -1dx + 2dx2

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

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

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

Add 'dxy' to each side of the equation.
-1dx + dxy + 1dy + dxy + -2dy2 = -1dxy + dxy + 2dx2

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

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

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

Add '-2dx2' to each side of the equation.
-1dx + 2dxy + 1dy + -2dx2 + -2dy2 = 2dx2 + -2dx2

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

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

Factor out the Greatest Common Factor (GCF), 'd'.
d(-1x + 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 '(-1x + 2xy + -2x2 + y + -2y2)' equal to zero and attempt to solve:

Simplifying
-1x + 2xy + -2x2 + y + -2y2 = 0

Solving
-1x + 2xy + -2x2 + y + -2y2 = 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplifying
0 = x + -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}