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

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


Simplifying
(2y + -1x) * dy + -1(4x + -3y) * dx = 0

Reorder the terms:
(-1x + 2y) * dy + -1(4x + -3y) * dx = 0

Reorder the terms for easier multiplication:
dy(-1x + 2y) + -1(4x + -3y) * dx = 0
(-1x * dy + 2y * dy) + -1(4x + -3y) * dx = 0
(-1dxy + 2dy2) + -1(4x + -3y) * dx = 0

Reorder the terms for easier multiplication:
-1dxy + 2dy2 + -1dx(4x + -3y) = 0
-1dxy + 2dy2 + (4x * -1dx + -3y * -1dx) = 0

Reorder the terms:
-1dxy + 2dy2 + (3dxy + -4dx2) = 0
-1dxy + 2dy2 + (3dxy + -4dx2) = 0

Reorder the terms:
-1dxy + 3dxy + -4dx2 + 2dy2 = 0

Combine like terms: -1dxy + 3dxy = 2dxy
2dxy + -4dx2 + 2dy2 = 0

Solving
2dxy + -4dx2 + 2dy2 = 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(xy + -2x2 + y2) = 0

Factor a trinomial.
2d((-2x + -1y)(x + -1y)) = 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 '(-2x + -1y)' equal to zero and attempt to solve:

Simplifying
-2x + -1y = 0

Solving
-2x + -1y = 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 + -1y = 0 + 2x

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

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

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

Simplifying
0 = 2x + y

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 + -1y)' equal to zero and attempt to solve:

Simplifying
x + -1y = 0

Solving
x + -1y = 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 + -1y = 0 + -1x

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

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

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

Simplifying
0 = -1x + y

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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