(2x-3y)dx+(5y-x)dy=0

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Solution for (2x-3y)dx+(5y-x)dy=0 equation: 


Simplifying
(2x + -3y) * dx + (5y + -1x) * dy = 0

Reorder the terms for easier multiplication:
dx(2x + -3y) + (5y + -1x) * dy = 0
(2x * dx + -3y * dx) + (5y + -1x) * dy = 0

Reorder the terms:
(-3dxy + 2dx2) + (5y + -1x) * dy = 0
(-3dxy + 2dx2) + (5y + -1x) * dy = 0

Reorder the terms:
-3dxy + 2dx2 + (-1x + 5y) * dy = 0

Reorder the terms for easier multiplication:
-3dxy + 2dx2 + dy(-1x + 5y) = 0
-3dxy + 2dx2 + (-1x * dy + 5y * dy) = 0
-3dxy + 2dx2 + (-1dxy + 5dy2) = 0

Reorder the terms:
-3dxy + -1dxy + 2dx2 + 5dy2 = 0

Combine like terms: -3dxy + -1dxy = -4dxy
-4dxy + 2dx2 + 5dy2 = 0

Solving
-4dxy + 2dx2 + 5dy2 = 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(-4xy + 2x2 + 5y2) = 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 '(-4xy + 2x2 + 5y2)' equal to zero and attempt to solve:

Simplifying
-4xy + 2x2 + 5y2 = 0

Solving
-4xy + 2x2 + 5y2 = 0

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

Add '4xy' to each side of the equation.
-4xy + 2x2 + 4xy + 5y2 = 0 + 4xy

Reorder the terms:
-4xy + 4xy + 2x2 + 5y2 = 0 + 4xy

Combine like terms: -4xy + 4xy = 0
0 + 2x2 + 5y2 = 0 + 4xy
2x2 + 5y2 = 0 + 4xy
Remove the zero:
2x2 + 5y2 = 4xy

Add '-2x2' to each side of the equation.
2x2 + -2x2 + 5y2 = 4xy + -2x2

Combine like terms: 2x2 + -2x2 = 0
0 + 5y2 = 4xy + -2x2
5y2 = 4xy + -2x2

Add '-5y2' to each side of the equation.
5y2 + -5y2 = 4xy + -2x2 + -5y2

Combine like terms: 5y2 + -5y2 = 0
0 = 4xy + -2x2 + -5y2

Simplifying
0 = 4xy + -2x2 + -5y2

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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