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

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

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

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

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

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

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

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

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

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

Combine like terms: dxy + 2dxy = 3dxy
-1dx + 3dxy + dx2 + 1dy + 2dy2 = 0

Solving
-1dx + 3dxy + dx2 + 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(-1x + 3xy + x2 + 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 + 3xy + x2 + y + 2y2)' equal to zero and attempt to solve:

Simplifying
-1x + 3xy + x2 + y + 2y2 = 0

Solving
-1x + 3xy + x2 + 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 + 3xy + x2 + y + x + 2y2 = 0 + x

Reorder the terms:
-1x + x + 3xy + x2 + y + 2y2 = 0 + x

Combine like terms: -1x + x = 0
0 + 3xy + x2 + y + 2y2 = 0 + x
3xy + x2 + y + 2y2 = 0 + x
Remove the zero:
3xy + x2 + y + 2y2 = x

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

Reorder the terms:
3xy + -3xy + x2 + y + 2y2 = x + -3xy

Combine like terms: 3xy + -3xy = 0
0 + x2 + y + 2y2 = x + -3xy
x2 + y + 2y2 = x + -3xy

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

Reorder the terms:
x2 + -1x2 + y + 2y2 = x + -3xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + y + 2y2 = x + -3xy + -1x2
y + 2y2 = x + -3xy + -1x2

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

Combine like terms: y + -1y = 0
0 + 2y2 = x + -3xy + -1x2 + -1y
2y2 = x + -3xy + -1x2 + -1y

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

Combine like terms: 2y2 + -2y2 = 0
0 = x + -3xy + -1x2 + -1y + -2y2

Simplifying
0 = x + -3xy + -1x2 + -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}