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

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


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

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

Multiply ins * dx
dinsx(x + y) + (2y + sin(x + y)) * dy = 0
(x * dinsx + y * dinsx) + (2y + sin(x + y)) * dy = 0

Reorder the terms:
(dinsxy + dinsx2) + (2y + sin(x + y)) * dy = 0
(dinsxy + dinsx2) + (2y + sin(x + y)) * dy = 0
dinsxy + dinsx2 + (2y + (x * ins + y * ins)) * dy = 0
dinsxy + dinsx2 + (2y + (insx + insy)) * dy = 0

Reorder the terms:
dinsxy + dinsx2 + (insx + insy + 2y) * dy = 0

Reorder the terms for easier multiplication:
dinsxy + dinsx2 + dy(insx + insy + 2y) = 0
dinsxy + dinsx2 + (insx * dy + insy * dy + 2y * dy) = 0
dinsxy + dinsx2 + (dinsxy + dinsy2 + 2dy2) = 0

Reorder the terms:
dinsxy + dinsxy + dinsx2 + dinsy2 + 2dy2 = 0

Combine like terms: dinsxy + dinsxy = 2dinsxy
2dinsxy + dinsx2 + dinsy2 + 2dy2 = 0

Solving
2dinsxy + dinsx2 + dinsy2 + 2dy2 = 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(2insxy + insx2 + insy2 + 2y2) = 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 '(2insxy + insx2 + insy2 + 2y2)' equal to zero and attempt to solve:

Simplifying
2insxy + insx2 + insy2 + 2y2 = 0

Solving
2insxy + insx2 + insy2 + 2y2 = 0

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

Add '-2insxy' to each side of the equation.
2insxy + insx2 + insy2 + -2insxy + 2y2 = 0 + -2insxy

Reorder the terms:
2insxy + -2insxy + insx2 + insy2 + 2y2 = 0 + -2insxy

Combine like terms: 2insxy + -2insxy = 0
0 + insx2 + insy2 + 2y2 = 0 + -2insxy
insx2 + insy2 + 2y2 = 0 + -2insxy
Remove the zero:
insx2 + insy2 + 2y2 = -2insxy

Add '-1insx2' to each side of the equation.
insx2 + insy2 + -1insx2 + 2y2 = -2insxy + -1insx2

Reorder the terms:
insx2 + -1insx2 + insy2 + 2y2 = -2insxy + -1insx2

Combine like terms: insx2 + -1insx2 = 0
0 + insy2 + 2y2 = -2insxy + -1insx2
insy2 + 2y2 = -2insxy + -1insx2

Add '-1insy2' to each side of the equation.
insy2 + -1insy2 + 2y2 = -2insxy + -1insx2 + -1insy2

Combine like terms: insy2 + -1insy2 = 0
0 + 2y2 = -2insxy + -1insx2 + -1insy2
2y2 = -2insxy + -1insx2 + -1insy2

Add '-2y2' to each side of the equation.
2y2 + -2y2 = -2insxy + -1insx2 + -1insy2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
0 = -2insxy + -1insx2 + -1insy2 + -2y2

Simplifying
0 = -2insxy + -1insx2 + -1insy2 + -2y2

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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