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

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


Simplifying
(x + 5y) * dy = -1(2x + 3y) * dx

Reorder the terms for easier multiplication:
dy(x + 5y) = -1(2x + 3y) * dx
(x * dy + 5y * dy) = -1(2x + 3y) * dx
(dxy + 5dy2) = -1(2x + 3y) * dx

Reorder the terms for easier multiplication:
dxy + 5dy2 = -1dx(2x + 3y)
dxy + 5dy2 = (2x * -1dx + 3y * -1dx)

Reorder the terms:
dxy + 5dy2 = (-3dxy + -2dx2)
dxy + 5dy2 = (-3dxy + -2dx2)

Solving
dxy + 5dy2 = -3dxy + -2dx2

Solving for variable 'd'.

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

Add '3dxy' to each side of the equation.
dxy + 3dxy + 5dy2 = -3dxy + 3dxy + -2dx2

Combine like terms: dxy + 3dxy = 4dxy
4dxy + 5dy2 = -3dxy + 3dxy + -2dx2

Combine like terms: -3dxy + 3dxy = 0
4dxy + 5dy2 = 0 + -2dx2
4dxy + 5dy2 = -2dx2

Add '2dx2' to each side of the equation.
4dxy + 2dx2 + 5dy2 = -2dx2 + 2dx2

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

Factor out the Greatest Common Factor (GCF), 'd'.
d(4xy + 2x2 + 5y2) = 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 '(4xy + 2x2 + 5y2)' equal to zero and attempt to solve:

Simplifying
4xy + 2x2 + 5y2 = 0

Solving
4xy + 2x2 + 5y2 = 0

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

Add '-4xy' to each side of the equation.
4xy + 2x2 + -4xy + 5y2 = 0 + -4xy

Reorder the terms:
4xy + -4xy + 2x2 + 5y2 = 0 + -4xy

Combine like terms: 4xy + -4xy = 0
0 + 2x2 + 5y2 = 0 + -4xy
2x2 + 5y2 = 0 + -4xy
Remove the zero:
2x2 + 5y2 = -4xy

Add '-2x2' to each side of the equation.
2x2 + -2x2 + 5y2 = -4xy + -2x2

Combine like terms: 2x2 + -2x2 = 0
0 + 5y2 = -4xy + -2x2
5y2 = -4xy + -2x2

Add '-5y2' to each side of the equation.
5y2 + -5y2 = -4xy + -2x2 + -5y2

Combine like terms: 5y2 + -5y2 = 0
0 = -4xy + -2x2 + -5y2

Simplifying
0 = -4xy + -2x2 + -5y2

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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