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

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


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

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

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

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

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

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

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

Simplifying
-1xy + x2 + y2 = 0

Solving
-1xy + x2 + y2 = 0

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

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

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

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

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

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

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

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

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

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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