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

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


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

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

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

Reorder the terms for easier multiplication:
-1dxy + dx2 + dy(3x + y) = 0
-1dxy + dx2 + (3x * dy + y * dy) = 0
-1dxy + dx2 + (3dxy + dy2) = 0

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

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

Solving
2dxy + 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(2xy + x2 + y2) = 0

Factor a trinomial.
d((x + y)(x + y)) = 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 + y)' equal to zero and attempt to solve:

Simplifying
x + y = 0

Solving
x + y = 0

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

Add '-1x' to each side of the equation.
x + -1x + y = 0 + -1x

Combine like terms: x + -1x = 0
0 + y = 0 + -1x
y = 0 + -1x
Remove the zero:
y = -1x

Add '-1y' to each side of the equation.
y + -1y = -1x + -1y

Combine like terms: y + -1y = 0
0 = -1x + -1y

Simplifying
0 = -1x + -1y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

Subproblem 3
Set the factor '(x + y)' equal to zero and attempt to solve:

Simplifying
x + y = 0

Solving
x + y = 0

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

Add '-1x' to each side of the equation.
x + -1x + y = 0 + -1x

Combine like terms: x + -1x = 0
0 + y = 0 + -1x
y = 0 + -1x
Remove the zero:
y = -1x

Add '-1y' to each side of the equation.
y + -1y = -1x + -1y

Combine like terms: y + -1y = 0
0 = -1x + -1y

Simplifying
0 = -1x + -1y

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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