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

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


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

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

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

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

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

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

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

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

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

Solving
1dx + 3dxy + dx2 + 2dy + 3dy2 = 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 + 3xy + x2 + 2y + 3y2) = 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 + 3xy + x2 + 2y + 3y2)' equal to zero and attempt to solve: Simplifying x + 3xy + x2 + 2y + 3y2 = 0 Solving x + 3xy + x2 + 2y + 3y2 = 0 Move all terms containing d to the left, all other terms to the right. Add '-1x' to each side of the equation. x + 3xy + x2 + 2y + -1x + 3y2 = 0 + -1x Reorder the terms: x + -1x + 3xy + x2 + 2y + 3y2 = 0 + -1x Combine like terms: x + -1x = 0 0 + 3xy + x2 + 2y + 3y2 = 0 + -1x 3xy + x2 + 2y + 3y2 = 0 + -1x Remove the zero: 3xy + x2 + 2y + 3y2 = -1x Add '-3xy' to each side of the equation. 3xy + x2 + 2y + -3xy + 3y2 = -1x + -3xy Reorder the terms: 3xy + -3xy + x2 + 2y + 3y2 = -1x + -3xy Combine like terms: 3xy + -3xy = 0 0 + x2 + 2y + 3y2 = -1x + -3xy x2 + 2y + 3y2 = -1x + -3xy Add '-1x2' to each side of the equation. x2 + 2y + -1x2 + 3y2 = -1x + -3xy + -1x2 Reorder the terms: x2 + -1x2 + 2y + 3y2 = -1x + -3xy + -1x2 Combine like terms: x2 + -1x2 = 0 0 + 2y + 3y2 = -1x + -3xy + -1x2 2y + 3y2 = -1x + -3xy + -1x2 Add '-2y' to each side of the equation. 2y + -2y + 3y2 = -1x + -3xy + -1x2 + -2y Combine like terms: 2y + -2y = 0 0 + 3y2 = -1x + -3xy + -1x2 + -2y 3y2 = -1x + -3xy + -1x2 + -2y Add '-3y2' to each side of the equation. 3y2 + -3y2 = -1x + -3xy + -1x2 + -2y + -3y2 Combine like terms: 3y2 + -3y2 = 0 0 = -1x + -3xy + -1x2 + -2y + -3y2 Simplifying 0 = -1x + -3xy + -1x2 + -2y + -3y2 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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