(x-5y)dx+(5x-y)dy=0

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Solution for (x-5y)dx+(5x-y)dy=0 equation: 


Simplifying
(x + -5y) * dx + (5x + -1y) * dy = 0

Reorder the terms for easier multiplication:
dx(x + -5y) + (5x + -1y) * dy = 0
(x * dx + -5y * dx) + (5x + -1y) * dy = 0

Reorder the terms:
(-5dxy + dx2) + (5x + -1y) * dy = 0
(-5dxy + dx2) + (5x + -1y) * dy = 0

Reorder the terms for easier multiplication:
-5dxy + dx2 + dy(5x + -1y) = 0
-5dxy + dx2 + (5x * dy + -1y * dy) = 0
-5dxy + dx2 + (5dxy + -1dy2) = 0

Reorder the terms:
-5dxy + 5dxy + dx2 + -1dy2 = 0

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

Solving
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(x2 + -1y2) = 0

Factor a difference between two squares.
d((x + y)(x + -1y)) = 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 '(x + y)' equal to zero and attempt to solve:

Simplifying
x + y = 0

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

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

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

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

Simplifying
0 = -1x + -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 '(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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