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

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


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

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

Reorder the terms:
(3dxy + dx2) + (2x + -1y) * dy = 0
(3dxy + dx2) + (2x + -1y) * dy = 0

Reorder the terms for easier multiplication:
3dxy + dx2 + dy(2x + -1y) = 0
3dxy + dx2 + (2x * dy + -1y * dy) = 0
3dxy + dx2 + (2dxy + -1dy2) = 0

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

Combine like terms: 3dxy + 2dxy = 5dxy
5dxy + dx2 + -1dy2 = 0

Solving
5dxy + dx2 + -1dy2 = 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 + x2 + -1y2) = 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 + x2 + -1y2)' equal to zero and attempt to solve:

Simplifying
5xy + x2 + -1y2 = 0

Solving
5xy + x2 + -1y2 = 0

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

Add '-5xy' to each side of the equation.
5xy + x2 + -5xy + -1y2 = 0 + -5xy

Reorder the terms:
5xy + -5xy + x2 + -1y2 = 0 + -5xy

Combine like terms: 5xy + -5xy = 0
0 + x2 + -1y2 = 0 + -5xy
x2 + -1y2 = 0 + -5xy
Remove the zero:
x2 + -1y2 = -5xy

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

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

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

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

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

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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