(x+2y)dx+(2x+3y+1)dy=0

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Solution for (x+2y)dx+(2x+3y+1)dy=0 equation: 


Simplifying
(x + 2y) * dx + (2x + 3y + 1) * dy = 0

Reorder the terms for easier multiplication:
dx(x + 2y) + (2x + 3y + 1) * dy = 0
(x * dx + 2y * dx) + (2x + 3y + 1) * dy = 0

Reorder the terms:
(2dxy + dx2) + (2x + 3y + 1) * dy = 0
(2dxy + dx2) + (2x + 3y + 1) * dy = 0

Reorder the terms:
2dxy + dx2 + (1 + 2x + 3y) * dy = 0

Reorder the terms for easier multiplication:
2dxy + dx2 + dy(1 + 2x + 3y) = 0
2dxy + dx2 + (1 * dy + 2x * dy + 3y * dy) = 0

Reorder the terms:
2dxy + dx2 + (2dxy + 1dy + 3dy2) = 0
2dxy + dx2 + (2dxy + 1dy + 3dy2) = 0

Reorder the terms:
2dxy + 2dxy + dx2 + 1dy + 3dy2 = 0

Combine like terms: 2dxy + 2dxy = 4dxy
4dxy + dx2 + 1dy + 3dy2 = 0

Solving
4dxy + dx2 + 1dy + 3dy2 = 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(4xy + x2 + y + 3y2) = 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 '(4xy + x2 + y + 3y2)' equal to zero and attempt to solve:

Simplifying
4xy + x2 + y + 3y2 = 0

Solving
4xy + x2 + y + 3y2 = 0

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

Add '-4xy' to each side of the equation.
4xy + x2 + y + -4xy + 3y2 = 0 + -4xy

Reorder the terms:
4xy + -4xy + x2 + y + 3y2 = 0 + -4xy

Combine like terms: 4xy + -4xy = 0
0 + x2 + y + 3y2 = 0 + -4xy
x2 + y + 3y2 = 0 + -4xy
Remove the zero:
x2 + y + 3y2 = -4xy

Add '-1x2' to each side of the equation.
x2 + y + -1x2 + 3y2 = -4xy + -1x2

Reorder the terms:
x2 + -1x2 + y + 3y2 = -4xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + y + 3y2 = -4xy + -1x2
y + 3y2 = -4xy + -1x2

Add '-1y' to each side of the equation.
y + -1y + 3y2 = -4xy + -1x2 + -1y

Combine like terms: y + -1y = 0
0 + 3y2 = -4xy + -1x2 + -1y
3y2 = -4xy + -1x2 + -1y

Add '-3y2' to each side of the equation.
3y2 + -3y2 = -4xy + -1x2 + -1y + -3y2

Combine like terms: 3y2 + -3y2 = 0
0 = -4xy + -1x2 + -1y + -3y2

Simplifying
0 = -4xy + -1x2 + -1y + -3y2

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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