(8x+4y+1)dx+(4x+2y+1)dy=0

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Solution for (8x+4y+1)dx+(4x+2y+1)dy=0 equation:


Simplifying
(8x + 4y + 1) * dx + (4x + 2y + 1) * dy = 0

Reorder the terms:
(1 + 8x + 4y) * dx + (4x + 2y + 1) * dy = 0

Reorder the terms for easier multiplication:
dx(1 + 8x + 4y) + (4x + 2y + 1) * dy = 0
(1 * dx + 8x * dx + 4y * dx) + (4x + 2y + 1) * dy = 0

Reorder the terms:
(1dx + 4dxy + 8dx2) + (4x + 2y + 1) * dy = 0
(1dx + 4dxy + 8dx2) + (4x + 2y + 1) * dy = 0

Reorder the terms:
1dx + 4dxy + 8dx2 + (1 + 4x + 2y) * dy = 0

Reorder the terms for easier multiplication:
1dx + 4dxy + 8dx2 + dy(1 + 4x + 2y) = 0
1dx + 4dxy + 8dx2 + (1 * dy + 4x * dy + 2y * dy) = 0

Reorder the terms:
1dx + 4dxy + 8dx2 + (4dxy + 1dy + 2dy2) = 0
1dx + 4dxy + 8dx2 + (4dxy + 1dy + 2dy2) = 0

Reorder the terms:
1dx + 4dxy + 4dxy + 8dx2 + 1dy + 2dy2 = 0

Combine like terms: 4dxy + 4dxy = 8dxy
1dx + 8dxy + 8dx2 + 1dy + 2dy2 = 0

Solving
1dx + 8dxy + 8dx2 + 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(x + 8xy + 8x2 + 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 '(x + 8xy + 8x2 + y + 2y2)' equal to zero and attempt to solve: Simplifying x + 8xy + 8x2 + y + 2y2 = 0 Solving x + 8xy + 8x2 + y + 2y2 = 0 Move all terms containing d to the left, all other terms to the right. Add '-1x' to each side of the equation. x + 8xy + 8x2 + y + -1x + 2y2 = 0 + -1x Reorder the terms: x + -1x + 8xy + 8x2 + y + 2y2 = 0 + -1x Combine like terms: x + -1x = 0 0 + 8xy + 8x2 + y + 2y2 = 0 + -1x 8xy + 8x2 + y + 2y2 = 0 + -1x Remove the zero: 8xy + 8x2 + y + 2y2 = -1x Add '-8xy' to each side of the equation. 8xy + 8x2 + y + -8xy + 2y2 = -1x + -8xy Reorder the terms: 8xy + -8xy + 8x2 + y + 2y2 = -1x + -8xy Combine like terms: 8xy + -8xy = 0 0 + 8x2 + y + 2y2 = -1x + -8xy 8x2 + y + 2y2 = -1x + -8xy Add '-8x2' to each side of the equation. 8x2 + y + -8x2 + 2y2 = -1x + -8xy + -8x2 Reorder the terms: 8x2 + -8x2 + y + 2y2 = -1x + -8xy + -8x2 Combine like terms: 8x2 + -8x2 = 0 0 + y + 2y2 = -1x + -8xy + -8x2 y + 2y2 = -1x + -8xy + -8x2 Add '-1y' to each side of the equation. y + -1y + 2y2 = -1x + -8xy + -8x2 + -1y Combine like terms: y + -1y = 0 0 + 2y2 = -1x + -8xy + -8x2 + -1y 2y2 = -1x + -8xy + -8x2 + -1y Add '-2y2' to each side of the equation. 2y2 + -2y2 = -1x + -8xy + -8x2 + -1y + -2y2 Combine like terms: 2y2 + -2y2 = 0 0 = -1x + -8xy + -8x2 + -1y + -2y2 Simplifying 0 = -1x + -8xy + -8x2 + -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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