(xlny+xy)dx+(ylnx+xy)xy=0

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Solution for (xlny+xy)dx+(ylnx+xy)xy=0 equation: 


Simplifying
(xlny + xy) * dx + (ylnx + xy) * xy = 0

Reorder the terms for easier multiplication:
dx(lnxy + xy) + (ylnx + xy) * xy = 0
(lnxy * dx + xy * dx) + (ylnx + xy) * xy = 0
(dlnx2y + dx2y) + (ylnx + xy) * xy = 0

Reorder the terms for easier multiplication:
dlnx2y + dx2y + xy(lnxy + xy) = 0
dlnx2y + dx2y + (lnxy * xy + xy * xy) = 0
dlnx2y + dx2y + (lnx2y2 + x2y2) = 0

Solving
dlnx2y + dx2y + lnx2y2 + x2y2 = 0

Solving for variable 'd'.

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

Add '-1lnx2y2' to each side of the equation.
dlnx2y + dx2y + lnx2y2 + -1lnx2y2 + x2y2 = 0 + -1lnx2y2

Combine like terms: lnx2y2 + -1lnx2y2 = 0
dlnx2y + dx2y + 0 + x2y2 = 0 + -1lnx2y2
dlnx2y + dx2y + x2y2 = 0 + -1lnx2y2
Remove the zero:
dlnx2y + dx2y + x2y2 = -1lnx2y2

Add '-1x2y2' to each side of the equation.
dlnx2y + dx2y + x2y2 + -1x2y2 = -1lnx2y2 + -1x2y2

Combine like terms: x2y2 + -1x2y2 = 0
dlnx2y + dx2y + 0 = -1lnx2y2 + -1x2y2
dlnx2y + dx2y = -1lnx2y2 + -1x2y2

Reorder the terms:
dlnx2y + dx2y + lnx2y2 + x2y2 = -1lnx2y2 + lnx2y2 + -1x2y2 + x2y2

Combine like terms: -1lnx2y2 + lnx2y2 = 0
dlnx2y + dx2y + lnx2y2 + x2y2 = 0 + -1x2y2 + x2y2
dlnx2y + dx2y + lnx2y2 + x2y2 = -1x2y2 + x2y2

Combine like terms: -1x2y2 + x2y2 = 0
dlnx2y + dx2y + lnx2y2 + x2y2 = 0

Factor out the Greatest Common Factor (GCF), 'x2y'.
x2y(dln + d + lny + y) = 0


Subproblem 1
Set the factor 'x2y' equal to zero and attempt to solve:

Simplifying
x2y = 0

Solving
x2y = 0

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

Add '-1x2y' to each side of the equation.
x2y + -1x2y = 0 + -1x2y
Remove the zero:
0 = -1x2y

Simplifying
0 = -1x2y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

Subproblem 2
Set the factor '(dln + d + lny + y)' equal to zero and attempt to solve:

Simplifying
dln + d + lny + y = 0

Reorder the terms:
d + dln + lny + y = 0

Solving
d + dln + lny + y = 0

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

Add '-1lny' to each side of the equation.
d + dln + lny + -1lny + y = 0 + -1lny

Combine like terms: lny + -1lny = 0
d + dln + 0 + y = 0 + -1lny
d + dln + y = 0 + -1lny
Remove the zero:
d + dln + y = -1lny

Add '-1y' to each side of the equation.
d + dln + y + -1y = -1lny + -1y

Combine like terms: y + -1y = 0
d + dln + 0 = -1lny + -1y
d + dln = -1lny + -1y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

The solution to this equation could not be determined.

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