(4x^2+8x^2y+4x^2y^2)dx+(x^2y+2y+3y^2)dy=0

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Solution for (4x^2+8x^2y+4x^2y^2)dx+(x^2y+2y+3y^2)dy=0 equation:


Simplifying
(4x2 + 8x2y + 4x2y2) * dx + (x2y + 2y + 3y2) * dy = 0

Reorder the terms for easier multiplication:
dx(4x2 + 8x2y + 4x2y2) + (x2y + 2y + 3y2) * dy = 0
(4x2 * dx + 8x2y * dx + 4x2y2 * dx) + (x2y + 2y + 3y2) * dy = 0
(4dx3 + 8dx3y + 4dx3y2) + (x2y + 2y + 3y2) * dy = 0

Reorder the terms for easier multiplication:
4dx3 + 8dx3y + 4dx3y2 + dy(x2y + 2y + 3y2) = 0
4dx3 + 8dx3y + 4dx3y2 + (x2y * dy + 2y * dy + 3y2 * dy) = 0
4dx3 + 8dx3y + 4dx3y2 + (dx2y2 + 2dy2 + 3dy3) = 0

Reorder the terms:
dx2y2 + 4dx3 + 8dx3y + 4dx3y2 + 2dy2 + 3dy3 = 0

Solving
dx2y2 + 4dx3 + 8dx3y + 4dx3y2 + 2dy2 + 3dy3 = 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(x2y2 + 4x3 + 8x3y + 4x3y2 + 2y2 + 3y3) = 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 '(x2y2 + 4x3 + 8x3y + 4x3y2 + 2y2 + 3y3)' equal to zero and attempt to solve: Simplifying x2y2 + 4x3 + 8x3y + 4x3y2 + 2y2 + 3y3 = 0 Solving x2y2 + 4x3 + 8x3y + 4x3y2 + 2y2 + 3y3 = 0 Move all terms containing d to the left, all other terms to the right. Add '-1x2y2' to each side of the equation. x2y2 + 4x3 + 8x3y + 4x3y2 + 2y2 + -1x2y2 + 3y3 = 0 + -1x2y2 Reorder the terms: x2y2 + -1x2y2 + 4x3 + 8x3y + 4x3y2 + 2y2 + 3y3 = 0 + -1x2y2 Combine like terms: x2y2 + -1x2y2 = 0 0 + 4x3 + 8x3y + 4x3y2 + 2y2 + 3y3 = 0 + -1x2y2 4x3 + 8x3y + 4x3y2 + 2y2 + 3y3 = 0 + -1x2y2 Remove the zero: 4x3 + 8x3y + 4x3y2 + 2y2 + 3y3 = -1x2y2 Add '-4x3' to each side of the equation. 4x3 + 8x3y + 4x3y2 + 2y2 + -4x3 + 3y3 = -1x2y2 + -4x3 Reorder the terms: 4x3 + -4x3 + 8x3y + 4x3y2 + 2y2 + 3y3 = -1x2y2 + -4x3 Combine like terms: 4x3 + -4x3 = 0 0 + 8x3y + 4x3y2 + 2y2 + 3y3 = -1x2y2 + -4x3 8x3y + 4x3y2 + 2y2 + 3y3 = -1x2y2 + -4x3 Add '-8x3y' to each side of the equation. 8x3y + 4x3y2 + 2y2 + -8x3y + 3y3 = -1x2y2 + -4x3 + -8x3y Reorder the terms: 8x3y + -8x3y + 4x3y2 + 2y2 + 3y3 = -1x2y2 + -4x3 + -8x3y Combine like terms: 8x3y + -8x3y = 0 0 + 4x3y2 + 2y2 + 3y3 = -1x2y2 + -4x3 + -8x3y 4x3y2 + 2y2 + 3y3 = -1x2y2 + -4x3 + -8x3y Add '-4x3y2' to each side of the equation. 4x3y2 + 2y2 + -4x3y2 + 3y3 = -1x2y2 + -4x3 + -8x3y + -4x3y2 Reorder the terms: 4x3y2 + -4x3y2 + 2y2 + 3y3 = -1x2y2 + -4x3 + -8x3y + -4x3y2 Combine like terms: 4x3y2 + -4x3y2 = 0 0 + 2y2 + 3y3 = -1x2y2 + -4x3 + -8x3y + -4x3y2 2y2 + 3y3 = -1x2y2 + -4x3 + -8x3y + -4x3y2 Add '-2y2' to each side of the equation. 2y2 + -2y2 + 3y3 = -1x2y2 + -4x3 + -8x3y + -4x3y2 + -2y2 Combine like terms: 2y2 + -2y2 = 0 0 + 3y3 = -1x2y2 + -4x3 + -8x3y + -4x3y2 + -2y2 3y3 = -1x2y2 + -4x3 + -8x3y + -4x3y2 + -2y2 Add '-3y3' to each side of the equation. 3y3 + -3y3 = -1x2y2 + -4x3 + -8x3y + -4x3y2 + -2y2 + -3y3 Combine like terms: 3y3 + -3y3 = 0 0 = -1x2y2 + -4x3 + -8x3y + -4x3y2 + -2y2 + -3y3 Simplifying 0 = -1x2y2 + -4x3 + -8x3y + -4x3y2 + -2y2 + -3y3 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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