(2x+3y)dx+(3x-4y)dy=0

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Solution for (2x+3y)dx+(3x-4y)dy=0 equation: 


Simplifying
(2x + 3y) * dx + (3x + -4y) * dy = 0

Reorder the terms for easier multiplication:
dx(2x + 3y) + (3x + -4y) * dy = 0
(2x * dx + 3y * dx) + (3x + -4y) * dy = 0

Reorder the terms:
(3dxy + 2dx2) + (3x + -4y) * dy = 0
(3dxy + 2dx2) + (3x + -4y) * dy = 0

Reorder the terms for easier multiplication:
3dxy + 2dx2 + dy(3x + -4y) = 0
3dxy + 2dx2 + (3x * dy + -4y * dy) = 0
3dxy + 2dx2 + (3dxy + -4dy2) = 0

Reorder the terms:
3dxy + 3dxy + 2dx2 + -4dy2 = 0

Combine like terms: 3dxy + 3dxy = 6dxy
6dxy + 2dx2 + -4dy2 = 0

Solving
6dxy + 2dx2 + -4dy2 = 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(3xy + x2 + -2y2) = 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 '(3xy + x2 + -2y2)' equal to zero and attempt to solve:

Simplifying
3xy + x2 + -2y2 = 0

Solving
3xy + x2 + -2y2 = 0

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

Add '-3xy' to each side of the equation.
3xy + x2 + -3xy + -2y2 = 0 + -3xy

Reorder the terms:
3xy + -3xy + x2 + -2y2 = 0 + -3xy

Combine like terms: 3xy + -3xy = 0
0 + x2 + -2y2 = 0 + -3xy
x2 + -2y2 = 0 + -3xy
Remove the zero:
x2 + -2y2 = -3xy

Add '-1x2' to each side of the equation.
x2 + -1x2 + -2y2 = -3xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + -2y2 = -3xy + -1x2
-2y2 = -3xy + -1x2

Add '2y2' to each side of the equation.
-2y2 + 2y2 = -3xy + -1x2 + 2y2

Combine like terms: -2y2 + 2y2 = 0
0 = -3xy + -1x2 + 2y2

Simplifying
0 = -3xy + -1x2 + 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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