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

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


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

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

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

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

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

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

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

Solving
2dxy + 4dx2 + -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 + 2x2 + -1y2) = 0

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

Simplifying
2x + -1y = 0

Solving
2x + -1y = 0

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

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

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

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

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

Simplifying
0 = -2x + y

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 + y)' equal to zero and attempt to solve:

Simplifying
x + y = 0

Solving
x + y = 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 + y = 0 + -1x

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

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

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

Simplifying
0 = -1x + -1y

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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