(3x+y)dy+(4x-y)dx=0

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Solution for (3x+y)dy+(4x-y)dx=0 equation: 


Simplifying
(3x + y) * dy + (4x + -1y) * dx = 0

Reorder the terms for easier multiplication:
dy(3x + y) + (4x + -1y) * dx = 0
(3x * dy + y * dy) + (4x + -1y) * dx = 0
(3dxy + dy2) + (4x + -1y) * dx = 0

Reorder the terms for easier multiplication:
3dxy + dy2 + dx(4x + -1y) = 0
3dxy + dy2 + (4x * dx + -1y * dx) = 0

Reorder the terms:
3dxy + dy2 + (-1dxy + 4dx2) = 0
3dxy + dy2 + (-1dxy + 4dx2) = 0

Reorder the terms:
3dxy + -1dxy + 4dx2 + dy2 = 0

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

Solving
2dxy + 4dx2 + 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(2xy + 4x2 + 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 '(2xy + 4x2 + y2)' equal to zero and attempt to solve:

Simplifying
2xy + 4x2 + y2 = 0

Solving
2xy + 4x2 + y2 = 0

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

Add '-2xy' to each side of the equation.
2xy + 4x2 + -2xy + y2 = 0 + -2xy

Reorder the terms:
2xy + -2xy + 4x2 + y2 = 0 + -2xy

Combine like terms: 2xy + -2xy = 0
0 + 4x2 + y2 = 0 + -2xy
4x2 + y2 = 0 + -2xy
Remove the zero:
4x2 + y2 = -2xy

Add '-4x2' to each side of the equation.
4x2 + -4x2 + y2 = -2xy + -4x2

Combine like terms: 4x2 + -4x2 = 0
0 + y2 = -2xy + -4x2
y2 = -2xy + -4x2

Add '-1y2' to each side of the equation.
y2 + -1y2 = -2xy + -4x2 + -1y2

Combine like terms: y2 + -1y2 = 0
0 = -2xy + -4x2 + -1y2

Simplifying
0 = -2xy + -4x2 + -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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