(6x+y)dx+(4x+y)dy=0

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Solution for (6x+y)dx+(4x+y)dy=0 equation: 


Simplifying
(6x + y) * dx + (4x + y) * dy = 0

Reorder the terms for easier multiplication:
dx(6x + y) + (4x + y) * dy = 0
(6x * dx + y * dx) + (4x + y) * dy = 0

Reorder the terms:
(dxy + 6dx2) + (4x + y) * dy = 0
(dxy + 6dx2) + (4x + y) * dy = 0

Reorder the terms for easier multiplication:
dxy + 6dx2 + dy(4x + y) = 0
dxy + 6dx2 + (4x * dy + y * dy) = 0
dxy + 6dx2 + (4dxy + dy2) = 0

Reorder the terms:
dxy + 4dxy + 6dx2 + dy2 = 0

Combine like terms: dxy + 4dxy = 5dxy
5dxy + 6dx2 + dy2 = 0

Solving
5dxy + 6dx2 + 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(5xy + 6x2 + y2) = 0

Factor a trinomial.
d((3x + y)(2x + 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 '(3x + y)' equal to zero and attempt to solve:

Simplifying
3x + y = 0

Solving
3x + y = 0

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

Add '-3x' to each side of the equation.
3x + -3x + y = 0 + -3x

Combine like terms: 3x + -3x = 0
0 + y = 0 + -3x
y = 0 + -3x
Remove the zero:
y = -3x

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

Combine like terms: y + -1y = 0
0 = -3x + -1y

Simplifying
0 = -3x + -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 '(2x + y)' equal to zero and attempt to solve:

Simplifying
2x + y = 0

Solving
2x + y = 0

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

Add '-2x' to each side of the equation.
2x + -2x + y = 0 + -2x

Combine like terms: 2x + -2x = 0
0 + y = 0 + -2x
y = 0 + -2x
Remove the zero:
y = -2x

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

Combine like terms: y + -1y = 0
0 = -2x + -1y

Simplifying
0 = -2x + -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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