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

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


Simplifying
x * dx + y * dy + 4y3(x2 + y2) * dy = 0

Multiply x * dx
dx2 + y * dy + 4y3(x2 + y2) * dy = 0

Multiply y * dy
dx2 + dy2 + 4y3(x2 + y2) * dy = 0

Reorder the terms for easier multiplication:
dx2 + dy2 + 4y3 * dy(x2 + y2) = 0

Multiply y3 * dy
dx2 + dy2 + 4dy4(x2 + y2) = 0
dx2 + dy2 + (x2 * 4dy4 + y2 * 4dy4) = 0
dx2 + dy2 + (4dx2y4 + 4dy6) = 0

Reorder the terms:
dx2 + 4dx2y4 + dy2 + 4dy6 = 0

Solving
dx2 + 4dx2y4 + dy2 + 4dy6 = 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(x2 + 4x2y4 + y2 + 4y6) = 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 '(x2 + 4x2y4 + y2 + 4y6)' equal to zero and attempt to solve: Simplifying x2 + 4x2y4 + y2 + 4y6 = 0 Solving x2 + 4x2y4 + y2 + 4y6 = 0 Move all terms containing d to the left, all other terms to the right. Add '-1x2' to each side of the equation. x2 + 4x2y4 + y2 + -1x2 + 4y6 = 0 + -1x2 Reorder the terms: x2 + -1x2 + 4x2y4 + y2 + 4y6 = 0 + -1x2 Combine like terms: x2 + -1x2 = 0 0 + 4x2y4 + y2 + 4y6 = 0 + -1x2 4x2y4 + y2 + 4y6 = 0 + -1x2 Remove the zero: 4x2y4 + y2 + 4y6 = -1x2 Add '-4x2y4' to each side of the equation. 4x2y4 + y2 + -4x2y4 + 4y6 = -1x2 + -4x2y4 Reorder the terms: 4x2y4 + -4x2y4 + y2 + 4y6 = -1x2 + -4x2y4 Combine like terms: 4x2y4 + -4x2y4 = 0 0 + y2 + 4y6 = -1x2 + -4x2y4 y2 + 4y6 = -1x2 + -4x2y4 Add '-1y2' to each side of the equation. y2 + -1y2 + 4y6 = -1x2 + -4x2y4 + -1y2 Combine like terms: y2 + -1y2 = 0 0 + 4y6 = -1x2 + -4x2y4 + -1y2 4y6 = -1x2 + -4x2y4 + -1y2 Add '-4y6' to each side of the equation. 4y6 + -4y6 = -1x2 + -4x2y4 + -1y2 + -4y6 Combine like terms: 4y6 + -4y6 = 0 0 = -1x2 + -4x2y4 + -1y2 + -4y6 Simplifying 0 = -1x2 + -4x2y4 + -1y2 + -4y6 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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