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

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


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

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

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

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

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

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

Solving
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(x2 + 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 '(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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