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

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

Simplifying
(x + 2y) * dx + (y + -1x) * dy = 0

Reorder the terms for easier multiplication:
dx(x + 2y) + (y + -1x) * dy = 0
(x * dx + 2y * dx) + (y + -1x) * dy = 0

Reorder the terms:
(2dxy + dx2) + (y + -1x) * dy = 0
(2dxy + dx2) + (y + -1x) * dy = 0

Reorder the terms:
2dxy + dx2 + (-1x + y) * dy = 0

Reorder the terms for easier multiplication:
2dxy + dx2 + dy(-1x + y) = 0
2dxy + dx2 + (-1x * dy + y * dy) = 0
2dxy + dx2 + (-1dxy + dy2) = 0

Reorder the terms:
2dxy + -1dxy + dx2 + dy2 = 0

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

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

Simplifying
xy + x2 + y2 = 0

Solving
xy + x2 + y2 = 0

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

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

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

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

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

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

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

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

Simplifying
0 = -1xy + -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}