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

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


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

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

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

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

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

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

Solving
dx2 + 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), 'd'.
d(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 '(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}

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