2(2x^2+y^2)dx+(2xdy+y)dy=0

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


Simplifying
2(2x2 + y2) * dx + (2xdy + y) * dy = 0

Reorder the terms for easier multiplication:
2dx(2x2 + y2) + (2xdy + y) * dy = 0
(2x2 * 2dx + y2 * 2dx) + (2xdy + y) * dy = 0

Reorder the terms:
(2dxy2 + 4dx3) + (2xdy + y) * dy = 0
(2dxy2 + 4dx3) + (2xdy + y) * dy = 0

Reorder the terms for easier multiplication:
2dxy2 + 4dx3 + dy(2dxy + y) = 0
2dxy2 + 4dx3 + (2dxy * dy + y * dy) = 0

Reorder the terms:
2dxy2 + 4dx3 + (dy2 + 2d2xy2) = 0
2dxy2 + 4dx3 + (dy2 + 2d2xy2) = 0

Solving
2dxy2 + 4dx3 + dy2 + 2d2xy2 = 0

Solving for variable 'd'.

Factor out the Greatest Common Factor (GCF), 'd'.
d(2xy2 + 4x3 + y2 + 2dxy2) = 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 '(2xy2 + 4x3 + y2 + 2dxy2)' equal to zero and attempt to solve:

Simplifying
2xy2 + 4x3 + y2 + 2dxy2 = 0

Reorder the terms:
2dxy2 + 2xy2 + 4x3 + y2 = 0

Solving
2dxy2 + 2xy2 + 4x3 + y2 = 0

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

Add '-2xy2' to each side of the equation.
2dxy2 + 2xy2 + 4x3 + -2xy2 + y2 = 0 + -2xy2

Reorder the terms:
2dxy2 + 2xy2 + -2xy2 + 4x3 + y2 = 0 + -2xy2

Combine like terms: 2xy2 + -2xy2 = 0
2dxy2 + 0 + 4x3 + y2 = 0 + -2xy2
2dxy2 + 4x3 + y2 = 0 + -2xy2
Remove the zero:
2dxy2 + 4x3 + y2 = -2xy2

Add '-4x3' to each side of the equation.
2dxy2 + 4x3 + -4x3 + y2 = -2xy2 + -4x3

Combine like terms: 4x3 + -4x3 = 0
2dxy2 + 0 + y2 = -2xy2 + -4x3
2dxy2 + y2 = -2xy2 + -4x3

Add '-1y2' to each side of the equation.
2dxy2 + y2 + -1y2 = -2xy2 + -4x3 + -1y2

Combine like terms: y2 + -1y2 = 0
2dxy2 + 0 = -2xy2 + -4x3 + -1y2
2dxy2 = -2xy2 + -4x3 + -1y2

Divide each side by '2xy2'.
d = -1 + -2x2y-2 + -0.5x-1

Simplifying
d = -1 + -2x2y-2 + -0.5x-1

Reorder the terms:
d = -1 + -0.5x-1 + -2x2y-2
Solution
d = {0, -1 + -0.5x-1 + -2x2y-2}

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