(4x-y)dx+(6y-x)dy=0

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Solution for (4x-y)dx+(6y-x)dy=0 equation:


Simplifying
(4x + -1y) * dx + (6y + -1x) * dy = 0

Reorder the terms for easier multiplication:
dx(4x + -1y) + (6y + -1x) * dy = 0
(4x * dx + -1y * dx) + (6y + -1x) * dy = 0

Reorder the terms:
(-1dxy + 4dx2) + (6y + -1x) * dy = 0
(-1dxy + 4dx2) + (6y + -1x) * dy = 0

Reorder the terms:
-1dxy + 4dx2 + (-1x + 6y) * dy = 0

Reorder the terms for easier multiplication:
-1dxy + 4dx2 + dy(-1x + 6y) = 0
-1dxy + 4dx2 + (-1x * dy + 6y * dy) = 0
-1dxy + 4dx2 + (-1dxy + 6dy2) = 0

Reorder the terms:
-1dxy + -1dxy + 4dx2 + 6dy2 = 0

Combine like terms: -1dxy + -1dxy = -2dxy
-2dxy + 4dx2 + 6dy2 = 0

Solving
-2dxy + 4dx2 + 6dy2 = 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), '2d'.
2d(-1xy + 2x2 + 3y2) = 0

Ignore the factor 2.

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