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

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

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

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

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

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

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

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

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

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

Solving
-1dx2 + -1dy2 = 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), '-1d'.
-1d(x2 + y2) = 0

Ignore the factor -1.
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 '(x2 + y2)' equal to zero and attempt to solve:

Simplifying
x2 + y2 = 0

Solving
x2 + y2 = 0

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

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

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

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

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

Simplifying
0 = -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}