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

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


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

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

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

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

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

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

Solving
4dxy + 2dx2 + -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), '2d'.
2d(2xy + x2 + -1y2) = 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 '(2xy + x2 + -1y2)' equal to zero and attempt to solve:

Simplifying
2xy + x2 + -1y2 = 0

Solving
2xy + x2 + -1y2 = 0

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

Add '-2xy' to each side of the equation.
2xy + x2 + -2xy + -1y2 = 0 + -2xy

Reorder the terms:
2xy + -2xy + x2 + -1y2 = 0 + -2xy

Combine like terms: 2xy + -2xy = 0
0 + x2 + -1y2 = 0 + -2xy
x2 + -1y2 = 0 + -2xy
Remove the zero:
x2 + -1y2 = -2xy

Add '-1x2' to each side of the equation.
x2 + -1x2 + -1y2 = -2xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + -1y2 = -2xy + -1x2
-1y2 = -2xy + -1x2

Add 'y2' to each side of the equation.
-1y2 + y2 = -2xy + -1x2 + y2

Combine like terms: -1y2 + y2 = 0
0 = -2xy + -1x2 + y2

Simplifying
0 = -2xy + -1x2 + y2

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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