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

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Solution for (3x+4y+1)dx+(4x+3y-1)dy=0 equation: 


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

Reorder the terms:
(1 + 3x + 4y) * dx + (4x + 3y + -1) * dy = 0

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

Reorder the terms:
(1dx + 4dxy + 3dx2) + (4x + 3y + -1) * dy = 0
(1dx + 4dxy + 3dx2) + (4x + 3y + -1) * dy = 0

Reorder the terms:
1dx + 4dxy + 3dx2 + (-1 + 4x + 3y) * dy = 0

Reorder the terms for easier multiplication:
1dx + 4dxy + 3dx2 + dy(-1 + 4x + 3y) = 0
1dx + 4dxy + 3dx2 + (-1 * dy + 4x * dy + 3y * dy) = 0

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

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

Combine like terms: 4dxy + 4dxy = 8dxy
1dx + 8dxy + 3dx2 + -1dy + 3dy2 = 0

Solving
1dx + 8dxy + 3dx2 + -1dy + 3dy2 = 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 + 8xy + 3x2 + -1y + 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 + 8xy + 3x2 + -1y + 3y2)' equal to zero and attempt to solve:

Simplifying
x + 8xy + 3x2 + -1y + 3y2 = 0

Solving
x + 8xy + 3x2 + -1y + 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 + 8xy + 3x2 + -1y + -1x + 3y2 = 0 + -1x

Reorder the terms:
x + -1x + 8xy + 3x2 + -1y + 3y2 = 0 + -1x

Combine like terms: x + -1x = 0
0 + 8xy + 3x2 + -1y + 3y2 = 0 + -1x
8xy + 3x2 + -1y + 3y2 = 0 + -1x
Remove the zero:
8xy + 3x2 + -1y + 3y2 = -1x

Add '-8xy' to each side of the equation.
8xy + 3x2 + -1y + -8xy + 3y2 = -1x + -8xy

Reorder the terms:
8xy + -8xy + 3x2 + -1y + 3y2 = -1x + -8xy

Combine like terms: 8xy + -8xy = 0
0 + 3x2 + -1y + 3y2 = -1x + -8xy
3x2 + -1y + 3y2 = -1x + -8xy

Add '-3x2' to each side of the equation.
3x2 + -1y + -3x2 + 3y2 = -1x + -8xy + -3x2

Reorder the terms:
3x2 + -3x2 + -1y + 3y2 = -1x + -8xy + -3x2

Combine like terms: 3x2 + -3x2 = 0
0 + -1y + 3y2 = -1x + -8xy + -3x2
-1y + 3y2 = -1x + -8xy + -3x2

Add 'y' to each side of the equation.
-1y + y + 3y2 = -1x + -8xy + -3x2 + y

Combine like terms: -1y + y = 0
0 + 3y2 = -1x + -8xy + -3x2 + y
3y2 = -1x + -8xy + -3x2 + y

Add '-3y2' to each side of the equation.
3y2 + -3y2 = -1x + -8xy + -3x2 + y + -3y2

Combine like terms: 3y2 + -3y2 = 0
0 = -1x + -8xy + -3x2 + y + -3y2

Simplifying
0 = -1x + -8xy + -3x2 + y + -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}

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