(2y-4x)dy+(y+x)dx=0

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Solution for (2y-4x)dy+(y+x)dx=0 equation: 


Simplifying
(2y + -4x) * dy + (y + x) * dx = 0

Reorder the terms:
(-4x + 2y) * dy + (y + x) * dx = 0

Reorder the terms for easier multiplication:
dy(-4x + 2y) + (y + x) * dx = 0
(-4x * dy + 2y * dy) + (y + x) * dx = 0
(-4dxy + 2dy2) + (y + x) * dx = 0

Reorder the terms:
-4dxy + 2dy2 + (x + y) * dx = 0

Reorder the terms for easier multiplication:
-4dxy + 2dy2 + dx(x + y) = 0
-4dxy + 2dy2 + (x * dx + y * dx) = 0

Reorder the terms:
-4dxy + 2dy2 + (dxy + dx2) = 0
-4dxy + 2dy2 + (dxy + dx2) = 0

Reorder the terms:
-4dxy + dxy + dx2 + 2dy2 = 0

Combine like terms: -4dxy + dxy = -3dxy
-3dxy + dx2 + 2dy2 = 0

Solving
-3dxy + dx2 + 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), 'd'.
d(-3xy + x2 + 2y2) = 0

Factor a trinomial.
d((x + -2y)(x + -1y)) = 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 '(x + -2y)' equal to zero and attempt to solve:

Simplifying
x + -2y = 0

Solving
x + -2y = 0

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

Add '-1x' to each side of the equation.
x + -1x + -2y = 0 + -1x

Combine like terms: x + -1x = 0
0 + -2y = 0 + -1x
-2y = 0 + -1x
Remove the zero:
-2y = -1x

Add '2y' to each side of the equation.
-2y + 2y = -1x + 2y

Combine like terms: -2y + 2y = 0
0 = -1x + 2y

Simplifying
0 = -1x + 2y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

Subproblem 3
Set the factor '(x + -1y)' equal to zero and attempt to solve:

Simplifying
x + -1y = 0

Solving
x + -1y = 0

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

Add '-1x' to each side of the equation.
x + -1x + -1y = 0 + -1x

Combine like terms: x + -1x = 0
0 + -1y = 0 + -1x
-1y = 0 + -1x
Remove the zero:
-1y = -1x

Add 'y' to each side of the equation.
-1y + y = -1x + y

Combine like terms: -1y + y = 0
0 = -1x + y

Simplifying
0 = -1x + y

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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