(16x+5y)dx+(3x+y)dy=0

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Solution for (16x+5y)dx+(3x+y)dy=0 equation:


Simplifying
(16x + 5y) * dx + (3x + y) * dy = 0

Reorder the terms for easier multiplication:
dx(16x + 5y) + (3x + y) * dy = 0
(16x * dx + 5y * dx) + (3x + y) * dy = 0

Reorder the terms:
(5dxy + 16dx2) + (3x + y) * dy = 0
(5dxy + 16dx2) + (3x + y) * dy = 0

Reorder the terms for easier multiplication:
5dxy + 16dx2 + dy(3x + y) = 0
5dxy + 16dx2 + (3x * dy + y * dy) = 0
5dxy + 16dx2 + (3dxy + dy2) = 0

Reorder the terms:
5dxy + 3dxy + 16dx2 + dy2 = 0

Combine like terms: 5dxy + 3dxy = 8dxy
8dxy + 16dx2 + dy2 = 0

Solving
8dxy + 16dx2 + dy2 = 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(8xy + 16x2 + y2) = 0

Factor a trinomial.
d((4x + y)(4x + 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 '(4x + y)' equal to zero and attempt to solve: Simplifying 4x + y = 0 Solving 4x + y = 0 Move all terms containing d to the left, all other terms to the right. Add '-4x' to each side of the equation. 4x + -4x + y = 0 + -4x Combine like terms: 4x + -4x = 0 0 + y = 0 + -4x y = 0 + -4x Remove the zero: y = -4x Add '-1y' to each side of the equation. y + -1y = -4x + -1y Combine like terms: y + -1y = 0 0 = -4x + -1y Simplifying 0 = -4x + -1y The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.

Subproblem 3

Set the factor '(4x + y)' equal to zero and attempt to solve: Simplifying 4x + y = 0 Solving 4x + y = 0 Move all terms containing d to the left, all other terms to the right. Add '-4x' to each side of the equation. 4x + -4x + y = 0 + -4x Combine like terms: 4x + -4x = 0 0 + y = 0 + -4x y = 0 + -4x Remove the zero: y = -4x Add '-1y' to each side of the equation. y + -1y = -4x + -1y Combine like terms: y + -1y = 0 0 = -4x + -1y Simplifying 0 = -4x + -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}

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