(x+1)dy+(4y-x)dx=0

Solution for (x+1)dy+(4y-x)dx=0 equation:

Simplifying
(x + 1) * dy + (4y + -1x) * dx = 0

Reorder the terms:
(1 + x) * dy + (4y + -1x) * dx = 0

Reorder the terms for easier multiplication:
dy(1 + x) + (4y + -1x) * dx = 0
(1 * dy + x * dy) + (4y + -1x) * dx = 0

Reorder the terms:
(dxy + 1dy) + (4y + -1x) * dx = 0
(dxy + 1dy) + (4y + -1x) * dx = 0

Reorder the terms:
dxy + 1dy + (-1x + 4y) * dx = 0

Reorder the terms for easier multiplication:
dxy + 1dy + dx(-1x + 4y) = 0
dxy + 1dy + (-1x * dx + 4y * dx) = 0

Reorder the terms:
dxy + 1dy + (4dxy + -1dx2) = 0
dxy + 1dy + (4dxy + -1dx2) = 0

Reorder the terms:
dxy + 4dxy + -1dx2 + 1dy = 0

Combine like terms: dxy + 4dxy = 5dxy
5dxy + -1dx2 + 1dy = 0

Solving
5dxy + -1dx2 + 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(5xy + -1x2 + 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 '(5xy + -1x2 + y)' equal to zero and attempt to solve:

Simplifying
5xy + -1x2 + y = 0

Solving
5xy + -1x2 + y = 0

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

Add '-5xy' to each side of the equation.
5xy + -1x2 + -5xy + y = 0 + -5xy

Reorder the terms:
5xy + -5xy + -1x2 + y = 0 + -5xy

Combine like terms: 5xy + -5xy = 0
0 + -1x2 + y = 0 + -5xy
-1x2 + y = 0 + -5xy
Remove the zero:
-1x2 + y = -5xy

Add 'x2' to each side of the equation.
-1x2 + x2 + y = -5xy + x2

Combine like terms: -1x2 + x2 = 0
0 + y = -5xy + x2
y = -5xy + x2

Add '-1y' to each side of the equation.
y + -1y = -5xy + x2 + -1y

Combine like terms: y + -1y = 0
0 = -5xy + x2 + -1y

Simplifying
0 = -5xy + x2 + -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}