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

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


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

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

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

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

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

Combine like terms: dxy + dxy = 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(xy + x2 + -1y2) = 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 '(xy + x2 + -1y2)' equal to zero and attempt to solve:

Simplifying
xy + x2 + -1y2 = 0

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

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

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

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

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

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

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

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

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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