(x+2y+1)dx+(2x+4y-5)dy=0

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Solution for (x+2y+1)dx+(2x+4y-5)dy=0 equation: 


Simplifying
(x + 2y + 1) * dx + (2x + 4y + -5) * dy = 0

Reorder the terms:
(1 + x + 2y) * dx + (2x + 4y + -5) * dy = 0

Reorder the terms for easier multiplication:
dx(1 + x + 2y) + (2x + 4y + -5) * dy = 0
(1 * dx + x * dx + 2y * dx) + (2x + 4y + -5) * dy = 0

Reorder the terms:
(1dx + 2dxy + dx2) + (2x + 4y + -5) * dy = 0
(1dx + 2dxy + dx2) + (2x + 4y + -5) * dy = 0

Reorder the terms:
1dx + 2dxy + dx2 + (-5 + 2x + 4y) * dy = 0

Reorder the terms for easier multiplication:
1dx + 2dxy + dx2 + dy(-5 + 2x + 4y) = 0
1dx + 2dxy + dx2 + (-5 * dy + 2x * dy + 4y * dy) = 0

Reorder the terms:
1dx + 2dxy + dx2 + (2dxy + -5dy + 4dy2) = 0
1dx + 2dxy + dx2 + (2dxy + -5dy + 4dy2) = 0

Reorder the terms:
1dx + 2dxy + 2dxy + dx2 + -5dy + 4dy2 = 0

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

Solving
1dx + 4dxy + dx2 + -5dy + 4dy2 = 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 + 4xy + x2 + -5y + 4y2) = 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 + 4xy + x2 + -5y + 4y2)' equal to zero and attempt to solve:

Simplifying
x + 4xy + x2 + -5y + 4y2 = 0

Solving
x + 4xy + x2 + -5y + 4y2 = 0

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

Add '-1x' to each side of the equation.
x + 4xy + x2 + -5y + -1x + 4y2 = 0 + -1x

Reorder the terms:
x + -1x + 4xy + x2 + -5y + 4y2 = 0 + -1x

Combine like terms: x + -1x = 0
0 + 4xy + x2 + -5y + 4y2 = 0 + -1x
4xy + x2 + -5y + 4y2 = 0 + -1x
Remove the zero:
4xy + x2 + -5y + 4y2 = -1x

Add '-4xy' to each side of the equation.
4xy + x2 + -5y + -4xy + 4y2 = -1x + -4xy

Reorder the terms:
4xy + -4xy + x2 + -5y + 4y2 = -1x + -4xy

Combine like terms: 4xy + -4xy = 0
0 + x2 + -5y + 4y2 = -1x + -4xy
x2 + -5y + 4y2 = -1x + -4xy

Add '-1x2' to each side of the equation.
x2 + -5y + -1x2 + 4y2 = -1x + -4xy + -1x2

Reorder the terms:
x2 + -1x2 + -5y + 4y2 = -1x + -4xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + -5y + 4y2 = -1x + -4xy + -1x2
-5y + 4y2 = -1x + -4xy + -1x2

Add '5y' to each side of the equation.
-5y + 5y + 4y2 = -1x + -4xy + -1x2 + 5y

Combine like terms: -5y + 5y = 0
0 + 4y2 = -1x + -4xy + -1x2 + 5y
4y2 = -1x + -4xy + -1x2 + 5y

Add '-4y2' to each side of the equation.
4y2 + -4y2 = -1x + -4xy + -1x2 + 5y + -4y2

Combine like terms: 4y2 + -4y2 = 0
0 = -1x + -4xy + -1x2 + 5y + -4y2

Simplifying
0 = -1x + -4xy + -1x2 + 5y + -4y2

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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