(4y+3x)dy+(y-2x)dx=0

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Solution for (4y+3x)dy+(y-2x)dx=0 equation: 


Simplifying
(4y + 3x) * dy + (y + -2x) * dx = 0

Reorder the terms:
(3x + 4y) * dy + (y + -2x) * dx = 0

Reorder the terms for easier multiplication:
dy(3x + 4y) + (y + -2x) * dx = 0
(3x * dy + 4y * dy) + (y + -2x) * dx = 0
(3dxy + 4dy2) + (y + -2x) * dx = 0

Reorder the terms:
3dxy + 4dy2 + (-2x + y) * dx = 0

Reorder the terms for easier multiplication:
3dxy + 4dy2 + dx(-2x + y) = 0
3dxy + 4dy2 + (-2x * dx + y * dx) = 0

Reorder the terms:
3dxy + 4dy2 + (dxy + -2dx2) = 0
3dxy + 4dy2 + (dxy + -2dx2) = 0

Reorder the terms:
3dxy + dxy + -2dx2 + 4dy2 = 0

Combine like terms: 3dxy + dxy = 4dxy
4dxy + -2dx2 + 4dy2 = 0

Solving
4dxy + -2dx2 + 4dy2 = 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), '2d'.
2d(2xy + -1x2 + 2y2) = 0

Ignore the factor 2.

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 '(2xy + -1x2 + 2y2)' equal to zero and attempt to solve:

Simplifying
2xy + -1x2 + 2y2 = 0

Solving
2xy + -1x2 + 2y2 = 0

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

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

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

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

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

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

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

Combine like terms: 2y2 + -2y2 = 0
0 = -2xy + x2 + -2y2

Simplifying
0 = -2xy + x2 + -2y2

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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