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

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


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

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

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

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

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

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

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

Solving
2dxy + -1dx2 + dy2 = 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(2xy + -1x2 + y2) = 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 '(2xy + -1x2 + y2)' equal to zero and attempt to solve:

Simplifying
2xy + -1x2 + y2 = 0

Solving
2xy + -1x2 + y2 = 0

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

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

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

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

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

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

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

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

Simplifying
0 = -2xy + x2 + -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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