(2x-3y-1)dx+(12x-18y-1)dy=0

Solution for (2x-3y-1)dx+(12x-18y-1)dy=0 equation:

Simplifying
(2x + -3y + -1) * dx + (12x + -18y + -1) * dy = 0

Reorder the terms:
(-1 + 2x + -3y) * dx + (12x + -18y + -1) * dy = 0

Reorder the terms for easier multiplication:
dx(-1 + 2x + -3y) + (12x + -18y + -1) * dy = 0
(-1 * dx + 2x * dx + -3y * dx) + (12x + -18y + -1) * dy = 0

Reorder the terms:
(-1dx + -3dxy + 2dx2) + (12x + -18y + -1) * dy = 0
(-1dx + -3dxy + 2dx2) + (12x + -18y + -1) * dy = 0

Reorder the terms:
-1dx + -3dxy + 2dx2 + (-1 + 12x + -18y) * dy = 0

Reorder the terms for easier multiplication:
-1dx + -3dxy + 2dx2 + dy(-1 + 12x + -18y) = 0
-1dx + -3dxy + 2dx2 + (-1 * dy + 12x * dy + -18y * dy) = 0

Reorder the terms:
-1dx + -3dxy + 2dx2 + (12dxy + -1dy + -18dy2) = 0
-1dx + -3dxy + 2dx2 + (12dxy + -1dy + -18dy2) = 0

Reorder the terms:
-1dx + -3dxy + 12dxy + 2dx2 + -1dy + -18dy2 = 0

Combine like terms: -3dxy + 12dxy = 9dxy
-1dx + 9dxy + 2dx2 + -1dy + -18dy2 = 0

Solving
-1dx + 9dxy + 2dx2 + -1dy + -18dy2 = 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 + 9xy + 2x2 + -1y + -18y2) = 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 + 9xy + 2x2 + -1y + -18y2)' equal to zero and attempt to solve:

Simplifying
-1x + 9xy + 2x2 + -1y + -18y2 = 0

Solving
-1x + 9xy + 2x2 + -1y + -18y2 = 0

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

Add 'x' to each side of the equation.
-1x + 9xy + 2x2 + -1y + x + -18y2 = 0 + x

Reorder the terms:
-1x + x + 9xy + 2x2 + -1y + -18y2 = 0 + x

Combine like terms: -1x + x = 0
0 + 9xy + 2x2 + -1y + -18y2 = 0 + x
9xy + 2x2 + -1y + -18y2 = 0 + x
Remove the zero:
9xy + 2x2 + -1y + -18y2 = x

Add '-9xy' to each side of the equation.
9xy + 2x2 + -1y + -9xy + -18y2 = x + -9xy

Reorder the terms:
9xy + -9xy + 2x2 + -1y + -18y2 = x + -9xy

Combine like terms: 9xy + -9xy = 0
0 + 2x2 + -1y + -18y2 = x + -9xy
2x2 + -1y + -18y2 = x + -9xy

Add '-2x2' to each side of the equation.
2x2 + -1y + -2x2 + -18y2 = x + -9xy + -2x2

Reorder the terms:
2x2 + -2x2 + -1y + -18y2 = x + -9xy + -2x2

Combine like terms: 2x2 + -2x2 = 0
0 + -1y + -18y2 = x + -9xy + -2x2
-1y + -18y2 = x + -9xy + -2x2

Add 'y' to each side of the equation.
-1y + y + -18y2 = x + -9xy + -2x2 + y

Combine like terms: -1y + y = 0
0 + -18y2 = x + -9xy + -2x2 + y
-18y2 = x + -9xy + -2x2 + y

Add '18y2' to each side of the equation.
-18y2 + 18y2 = x + -9xy + -2x2 + y + 18y2

Combine like terms: -18y2 + 18y2 = 0
0 = x + -9xy + -2x2 + y + 18y2

Simplifying
0 = x + -9xy + -2x2 + y + 18y2

The solution to this equation could not be determined.

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