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

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


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

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

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

Reorder the terms for easier multiplication:
3dxy + dx2 + dy(3x + y) = 0
3dxy + dx2 + (3x * dy + y * dy) = 0
3dxy + dx2 + (3dxy + dy2) = 0

Reorder the terms:
3dxy + 3dxy + dx2 + dy2 = 0

Combine like terms: 3dxy + 3dxy = 6dxy
6dxy + dx2 + dy2 = 0

Solving
6dxy + dx2 + 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(6xy + x2 + 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 '(6xy + x2 + y2)' equal to zero and attempt to solve:

Simplifying
6xy + x2 + y2 = 0

Solving
6xy + x2 + y2 = 0

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

Add '-6xy' to each side of the equation.
6xy + x2 + -6xy + y2 = 0 + -6xy

Reorder the terms:
6xy + -6xy + x2 + y2 = 0 + -6xy

Combine like terms: 6xy + -6xy = 0
0 + x2 + y2 = 0 + -6xy
x2 + y2 = 0 + -6xy
Remove the zero:
x2 + y2 = -6xy

Add '-1x2' to each side of the equation.
x2 + -1x2 + y2 = -6xy + -1x2

Combine like terms: x2 + -1x2 = 0
0 + y2 = -6xy + -1x2
y2 = -6xy + -1x2

Add '-1y2' to each side of the equation.
y2 + -1y2 = -6xy + -1x2 + -1y2

Combine like terms: y2 + -1y2 = 0
0 = -6xy + -1x2 + -1y2

Simplifying
0 = -6xy + -1x2 + -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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