(2x+y+1)dx+(4x+2y-1)dy=0

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Solution for (2x+y+1)dx+(4x+2y-1)dy=0 equation: 


Simplifying
(2x + y + 1) * dx + (4x + 2y + -1) * dy = 0

Reorder the terms:
(1 + 2x + y) * dx + (4x + 2y + -1) * dy = 0

Reorder the terms for easier multiplication:
dx(1 + 2x + y) + (4x + 2y + -1) * dy = 0
(1 * dx + 2x * dx + y * dx) + (4x + 2y + -1) * dy = 0

Reorder the terms:
(1dx + dxy + 2dx2) + (4x + 2y + -1) * dy = 0
(1dx + dxy + 2dx2) + (4x + 2y + -1) * dy = 0

Reorder the terms:
1dx + dxy + 2dx2 + (-1 + 4x + 2y) * dy = 0

Reorder the terms for easier multiplication:
1dx + dxy + 2dx2 + dy(-1 + 4x + 2y) = 0
1dx + dxy + 2dx2 + (-1 * dy + 4x * dy + 2y * dy) = 0

Reorder the terms:
1dx + dxy + 2dx2 + (4dxy + -1dy + 2dy2) = 0
1dx + dxy + 2dx2 + (4dxy + -1dy + 2dy2) = 0

Reorder the terms:
1dx + dxy + 4dxy + 2dx2 + -1dy + 2dy2 = 0

Combine like terms: dxy + 4dxy = 5dxy
1dx + 5dxy + 2dx2 + -1dy + 2dy2 = 0

Solving
1dx + 5dxy + 2dx2 + -1dy + 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(x + 5xy + 2x2 + -1y + 2y2) = 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 + 5xy + 2x2 + -1y + 2y2)' equal to zero and attempt to solve:

Simplifying
x + 5xy + 2x2 + -1y + 2y2 = 0

Solving
x + 5xy + 2x2 + -1y + 2y2 = 0

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

Add '-1x' to each side of the equation.
x + 5xy + 2x2 + -1y + -1x + 2y2 = 0 + -1x

Reorder the terms:
x + -1x + 5xy + 2x2 + -1y + 2y2 = 0 + -1x

Combine like terms: x + -1x = 0
0 + 5xy + 2x2 + -1y + 2y2 = 0 + -1x
5xy + 2x2 + -1y + 2y2 = 0 + -1x
Remove the zero:
5xy + 2x2 + -1y + 2y2 = -1x

Add '-5xy' to each side of the equation.
5xy + 2x2 + -1y + -5xy + 2y2 = -1x + -5xy

Reorder the terms:
5xy + -5xy + 2x2 + -1y + 2y2 = -1x + -5xy

Combine like terms: 5xy + -5xy = 0
0 + 2x2 + -1y + 2y2 = -1x + -5xy
2x2 + -1y + 2y2 = -1x + -5xy

Add '-2x2' to each side of the equation.
2x2 + -1y + -2x2 + 2y2 = -1x + -5xy + -2x2

Reorder the terms:
2x2 + -2x2 + -1y + 2y2 = -1x + -5xy + -2x2

Combine like terms: 2x2 + -2x2 = 0
0 + -1y + 2y2 = -1x + -5xy + -2x2
-1y + 2y2 = -1x + -5xy + -2x2

Add 'y' to each side of the equation.
-1y + y + 2y2 = -1x + -5xy + -2x2 + y

Combine like terms: -1y + y = 0
0 + 2y2 = -1x + -5xy + -2x2 + y
2y2 = -1x + -5xy + -2x2 + y

Add '-2y2' to each side of the equation.
2y2 + -2y2 = -1x + -5xy + -2x2 + y + -2y2

Combine like terms: 2y2 + -2y2 = 0
0 = -1x + -5xy + -2x2 + y + -2y2

Simplifying
0 = -1x + -5xy + -2x2 + y + -2y2

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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