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

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


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

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

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

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

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

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

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

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

Combine like terms: dxy + dxy = 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(-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}

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