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

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

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

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

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

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

Simplifying
xy + -1x2 + y2 = 0

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

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

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

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

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

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

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

Simplifying
0 = -1xy + 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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