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

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


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

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

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

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

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

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

Combine like terms: 4dxy + dxy = 5dxy
5dxy + -2dx2 + dy2 = 0

Solving
5dxy + -2dx2 + 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 + -2x2 + y2) = 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 '(5xy + -2x2 + y2)' equal to zero and attempt to solve:

Simplifying
5xy + -2x2 + y2 = 0

Solving
5xy + -2x2 + y2 = 0

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

Add '-5xy' to each side of the equation.
5xy + -2x2 + -5xy + y2 = 0 + -5xy

Reorder the terms:
5xy + -5xy + -2x2 + y2 = 0 + -5xy

Combine like terms: 5xy + -5xy = 0
0 + -2x2 + y2 = 0 + -5xy
-2x2 + y2 = 0 + -5xy
Remove the zero:
-2x2 + y2 = -5xy

Add '2x2' to each side of the equation.
-2x2 + 2x2 + y2 = -5xy + 2x2

Combine like terms: -2x2 + 2x2 = 0
0 + y2 = -5xy + 2x2
y2 = -5xy + 2x2

Add '-1y2' to each side of the equation.
y2 + -1y2 = -5xy + 2x2 + -1y2

Combine like terms: y2 + -1y2 = 0
0 = -5xy + 2x2 + -1y2

Simplifying
0 = -5xy + 2x2 + -1y2

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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