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

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


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

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

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

Reorder the terms for easier multiplication:
1dx + 2dxy + dy(3x + y) = 0
1dx + 2dxy + (3x * dy + y * dy) = 0
1dx + 2dxy + (3dxy + dy2) = 0

Combine like terms: 2dxy + 3dxy = 5dxy
1dx + 5dxy + dy2 = 0

Solving
1dx + 5dxy + dy2 = 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 + y2) = 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 + y2)' equal to zero and attempt to solve:

Simplifying
x + 5xy + y2 = 0

Solving
x + 5xy + y2 = 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 + -1x + y2 = 0 + -1x

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

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

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

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

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

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

Simplifying
0 = -1x + -5xy + -1y2

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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