(x+3y+1)dy=(x+3y-1)dx

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

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

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

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

Reorder the terms:
(dxy + 1dy + 3dy2) = (x + 3y + -1) * dx
(dxy + 1dy + 3dy2) = (x + 3y + -1) * dx

Reorder the terms:
dxy + 1dy + 3dy2 = (-1 + x + 3y) * dx

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

Reorder the terms:
dxy + 1dy + 3dy2 = (-1dx + 3dxy + dx2)
dxy + 1dy + 3dy2 = (-1dx + 3dxy + dx2)

Solving
dxy + 1dy + 3dy2 = -1dx + 3dxy + dx2

Solving for variable 'd'.

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

Add 'dx' to each side of the equation.
dxy + 1dy + dx + 3dy2 = -1dx + 3dxy + dx + dx2

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

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

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

Add '-3dxy' to each side of the equation.
dx + dxy + 1dy + -3dxy + 3dy2 = 3dxy + -3dxy + dx2

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

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

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

Add '-1dx2' to each side of the equation.
dx + -2dxy + 1dy + -1dx2 + 3dy2 = dx2 + -1dx2

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

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

Factor out the Greatest Common Factor (GCF), 'd'.
d(x + -2xy + -1x2 + 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 '(x + -2xy + -1x2 + y + 3y2)' equal to zero and attempt to solve:

Simplifying
x + -2xy + -1x2 + y + 3y2 = 0

Solving
x + -2xy + -1x2 + y + 3y2 = 0

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

Add '-1x' to each side of the equation.
x + -2xy + -1x2 + y + -1x + 3y2 = 0 + -1x

Reorder the terms:
x + -1x + -2xy + -1x2 + y + 3y2 = 0 + -1x

Combine like terms: x + -1x = 0
0 + -2xy + -1x2 + y + 3y2 = 0 + -1x
-2xy + -1x2 + y + 3y2 = 0 + -1x
Remove the zero:
-2xy + -1x2 + y + 3y2 = -1x

Add '2xy' to each side of the equation.
-2xy + -1x2 + y + 2xy + 3y2 = -1x + 2xy

Reorder the terms:
-2xy + 2xy + -1x2 + y + 3y2 = -1x + 2xy

Combine like terms: -2xy + 2xy = 0
0 + -1x2 + y + 3y2 = -1x + 2xy
-1x2 + y + 3y2 = -1x + 2xy

Add 'x2' to each side of the equation.
-1x2 + y + x2 + 3y2 = -1x + 2xy + x2

Reorder the terms:
-1x2 + x2 + y + 3y2 = -1x + 2xy + x2

Combine like terms: -1x2 + x2 = 0
0 + y + 3y2 = -1x + 2xy + x2
y + 3y2 = -1x + 2xy + x2

Add '-1y' to each side of the equation.
y + -1y + 3y2 = -1x + 2xy + x2 + -1y

Combine like terms: y + -1y = 0
0 + 3y2 = -1x + 2xy + x2 + -1y
3y2 = -1x + 2xy + x2 + -1y

Add '-3y2' to each side of the equation.
3y2 + -3y2 = -1x + 2xy + x2 + -1y + -3y2

Combine like terms: 3y2 + -3y2 = 0
0 = -1x + 2xy + x2 + -1y + -3y2

Simplifying
0 = -1x + 2xy + x2 + -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}