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

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


Simplifying
(4y + x) * dx + (4x + -1y) * dy = 0

Reorder the terms:
(x + 4y) * dx + (4x + -1y) * dy = 0

Reorder the terms for easier multiplication:
dx(x + 4y) + (4x + -1y) * dy = 0
(x * dx + 4y * dx) + (4x + -1y) * dy = 0

Reorder the terms:
(4dxy + dx2) + (4x + -1y) * dy = 0
(4dxy + dx2) + (4x + -1y) * dy = 0

Reorder the terms for easier multiplication:
4dxy + dx2 + dy(4x + -1y) = 0
4dxy + dx2 + (4x * dy + -1y * dy) = 0
4dxy + dx2 + (4dxy + -1dy2) = 0

Reorder the terms:
4dxy + 4dxy + dx2 + -1dy2 = 0

Combine like terms: 4dxy + 4dxy = 8dxy
8dxy + dx2 + -1dy2 = 0

Solving
8dxy + 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(8xy + x2 + -1y2) = 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 '(8xy + x2 + -1y2)' equal to zero and attempt to solve:

Simplifying
8xy + x2 + -1y2 = 0

Solving
8xy + x2 + -1y2 = 0

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

Add '-8xy' to each side of the equation.
8xy + x2 + -8xy + -1y2 = 0 + -8xy

Reorder the terms:
8xy + -8xy + x2 + -1y2 = 0 + -8xy

Combine like terms: 8xy + -8xy = 0
0 + x2 + -1y2 = 0 + -8xy
x2 + -1y2 = 0 + -8xy
Remove the zero:
x2 + -1y2 = -8xy

Add '-1x2' to each side of the equation.
x2 + -1x2 + -1y2 = -8xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + -1y2 = -8xy + -1x2
-1y2 = -8xy + -1x2

Add 'y2' to each side of the equation.
-1y2 + y2 = -8xy + -1x2 + y2

Combine like terms: -1y2 + y2 = 0
0 = -8xy + -1x2 + y2

Simplifying
0 = -8xy + -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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