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

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


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

Reorder the terms for easier multiplication:
dy(2x + 3y) + (y + -1x) * dx = 0
(2x * dy + 3y * dy) + (y + -1x) * dx = 0
(2dxy + 3dy2) + (y + -1x) * dx = 0

Reorder the terms:
2dxy + 3dy2 + (-1x + y) * dx = 0

Reorder the terms for easier multiplication:
2dxy + 3dy2 + dx(-1x + y) = 0
2dxy + 3dy2 + (-1x * dx + y * dx) = 0

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

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

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

Solving
3dxy + -1dx2 + 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(3xy + -1x2 + 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 '(3xy + -1x2 + 3y2)' equal to zero and attempt to solve:

Simplifying
3xy + -1x2 + 3y2 = 0

Solving
3xy + -1x2 + 3y2 = 0

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

Add '-3xy' to each side of the equation.
3xy + -1x2 + -3xy + 3y2 = 0 + -3xy

Reorder the terms:
3xy + -3xy + -1x2 + 3y2 = 0 + -3xy

Combine like terms: 3xy + -3xy = 0
0 + -1x2 + 3y2 = 0 + -3xy
-1x2 + 3y2 = 0 + -3xy
Remove the zero:
-1x2 + 3y2 = -3xy

Add 'x2' to each side of the equation.
-1x2 + x2 + 3y2 = -3xy + x2

Combine like terms: -1x2 + x2 = 0
0 + 3y2 = -3xy + x2
3y2 = -3xy + x2

Add '-3y2' to each side of the equation.
3y2 + -3y2 = -3xy + x2 + -3y2

Combine like terms: 3y2 + -3y2 = 0
0 = -3xy + x2 + -3y2

Simplifying
0 = -3xy + x2 + -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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