(2x+5y)dx+(5x+3y)dy=0

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Solution for (2x+5y)dx+(5x+3y)dy=0 equation: 


Simplifying
(2x + 5y) * dx + (5x + 3y) * dy = 0

Reorder the terms for easier multiplication:
dx(2x + 5y) + (5x + 3y) * dy = 0
(2x * dx + 5y * dx) + (5x + 3y) * dy = 0

Reorder the terms:
(5dxy + 2dx2) + (5x + 3y) * dy = 0
(5dxy + 2dx2) + (5x + 3y) * dy = 0

Reorder the terms for easier multiplication:
5dxy + 2dx2 + dy(5x + 3y) = 0
5dxy + 2dx2 + (5x * dy + 3y * dy) = 0
5dxy + 2dx2 + (5dxy + 3dy2) = 0

Reorder the terms:
5dxy + 5dxy + 2dx2 + 3dy2 = 0

Combine like terms: 5dxy + 5dxy = 10dxy
10dxy + 2dx2 + 3dy2 = 0

Solving
10dxy + 2dx2 + 3dy2 = 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(10xy + 2x2 + 3y2) = 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 '(10xy + 2x2 + 3y2)' equal to zero and attempt to solve:

Simplifying
10xy + 2x2 + 3y2 = 0

Solving
10xy + 2x2 + 3y2 = 0

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

Add '-10xy' to each side of the equation.
10xy + 2x2 + -10xy + 3y2 = 0 + -10xy

Reorder the terms:
10xy + -10xy + 2x2 + 3y2 = 0 + -10xy

Combine like terms: 10xy + -10xy = 0
0 + 2x2 + 3y2 = 0 + -10xy
2x2 + 3y2 = 0 + -10xy
Remove the zero:
2x2 + 3y2 = -10xy

Add '-2x2' to each side of the equation.
2x2 + -2x2 + 3y2 = -10xy + -2x2

Combine like terms: 2x2 + -2x2 = 0
0 + 3y2 = -10xy + -2x2
3y2 = -10xy + -2x2

Add '-3y2' to each side of the equation.
3y2 + -3y2 = -10xy + -2x2 + -3y2

Combine like terms: 3y2 + -3y2 = 0
0 = -10xy + -2x2 + -3y2

Simplifying
0 = -10xy + -2x2 + -3y2

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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