(x+1)dy+(y+2)dx=0

Solution for (x+1)dy+(y+2)dx=0 equation:

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

Reorder the terms:
(1 + x) * dy + (y + 2) * dx = 0

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

Reorder the terms:
(dxy + 1dy) + (y + 2) * dx = 0
(dxy + 1dy) + (y + 2) * dx = 0

Reorder the terms:
dxy + 1dy + (2 + y) * dx = 0

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

Reorder the terms:
2dx + dxy + dxy + 1dy = 0

Combine like terms: dxy + dxy = 2dxy
2dx + 2dxy + 1dy = 0

Solving
2dx + 2dxy + 1dy = 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(2x + 2xy + y) = 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 '(2x + 2xy + y)' equal to zero and attempt to solve:

Simplifying
2x + 2xy + y = 0

Solving
2x + 2xy + y = 0

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

Add '-2x' to each side of the equation.
2x + 2xy + -2x + y = 0 + -2x

Reorder the terms:
2x + -2x + 2xy + y = 0 + -2x

Combine like terms: 2x + -2x = 0
0 + 2xy + y = 0 + -2x
2xy + y = 0 + -2x
Remove the zero:
2xy + y = -2x

Add '-2xy' to each side of the equation.
2xy + -2xy + y = -2x + -2xy

Combine like terms: 2xy + -2xy = 0
0 + y = -2x + -2xy
y = -2x + -2xy

Add '-1y' to each side of the equation.
y + -1y = -2x + -2xy + -1y

Combine like terms: y + -1y = 0
0 = -2x + -2xy + -1y

Simplifying
0 = -2x + -2xy + -1y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
d = {0}