(3x+8)(y^2+4)dx-4y(x^2+5x+6)dy=0, y(1)=2
Answers
Answer:
solution
Step-by-step explanation:
Simplifying
(3x + 8)(y2 + 4) * dx + -4y(x2 + 5x + 6) * dy = 0
Reorder the terms:
(8 + 3x)(y2 + 4) * dx + -4y(x2 + 5x + 6) * dy = 0
Reorder the terms:
(8 + 3x)(4 + y2) * dx + -4y(x2 + 5x + 6) * dy = 0
Reorder the terms for easier multiplication:
dx(8 + 3x)(4 + y2) + -4y(x2 + 5x + 6) * dy = 0
Multiply (8 + 3x) * (4 + y2)
dx(8(4 + y2) + 3x * (4 + y2)) + -4y(x2 + 5x + 6) * dy = 0
dx((4 * 8 + y2 * 8) + 3x * (4 + y2)) + -4y(x2 + 5x + 6) * dy = 0
dx((32 + 8y2) + 3x * (4 + y2)) + -4y(x2 + 5x + 6) * dy = 0
dx(32 + 8y2 + (4 * 3x + y2 * 3x)) + -4y(x2 + 5x + 6) * dy = 0
dx(32 + 8y2 + (12x + 3xy2)) + -4y(x2 + 5x + 6) * dy = 0
Reorder the terms:
dx(32 + 12x + 3xy2 + 8y2) + -4y(x2 + 5x + 6) * dy = 0
dx(32 + 12x + 3xy2 + 8y2) + -4y(x2 + 5x + 6) * dy = 0
(32 * dx + 12x * dx + 3xy2 * dx + 8y2 * dx) + -4y(x2 + 5x + 6) * dy = 0
Reorder the terms:
(32dx + 8dxy2 + 12dx2 + 3dx2y2) + -4y(x2 + 5x + 6) * dy = 0
(32dx + 8dxy2 + 12dx2 + 3dx2y2) + -4y(x2 + 5x + 6) * dy = 0
Reorder the terms:
32dx + 8dxy2 + 12dx2 + 3dx2y2 + -4y(6 + 5x + x2) * dy = 0
Reorder the terms for easier multiplication:
32dx + 8dxy2 + 12dx2 + 3dx2y2 + -4y * dy(6 + 5x + x2) = 0
Multiply y * dy
32dx + 8dxy2 + 12dx2 + 3dx2y2 + -4dy2(6 + 5x + x2) = 0
32dx + 8dxy2 + 12dx2 + 3dx2y2 + (6 * -4dy2 + 5x * -4dy2 + x2 * -4dy2) = 0
Reorder the terms:
32dx + 8dxy2 + 12dx2 + 3dx2y2 + (-20dxy2 + -4dx2y2 + -24dy2) = 0
32dx + 8dxy2 + 12dx2 + 3dx2y2 + (-20dxy2 + -4dx2y2 + -24dy2) = 0
Reorder the terms:
32dx + 8dxy2 + -20dxy2 + 12dx2 + 3dx2y2 + -4dx2y2 + -24dy2 = 0
Combine like terms: 8dxy2 + -20dxy2 = -12dxy2
32dx + -12dxy2 + 12dx2 + 3dx2y2 + -4dx2y2 + -24dy2 = 0
Combine like terms: 3dx2y2 + -4dx2y2 = -1dx2y2
32dx + -12dxy2 + 12dx2 + -1dx2y2 + -24dy2 = 0
Solving
32dx + -12dxy2 + 12dx2 + -1dx2y2 + -24dy2 = 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(32x + -12xy2 + 12x2 + -1x2y2 + -24y2) = 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 '(32x + -12xy2 + 12x2 + -1x2y2 + -24y2)' equal to zero and attempt to solve:
Simplifying
32x + -12xy2 + 12x2 + -1x2y2 + -24y2 = 0
Solving
32x + -12xy2 + 12x2 + -1x2y2 + -24y2 = 0
Move all terms containing d to the left, all other terms to the right.
Add '-32x' to each side of the equation.
32x + -12xy2 + 12x2 + -1x2y2 + -32x + -24y2 = 0 + -32x
Reorder the terms:
32x + -32x + -12xy2 + 12x2 + -1x2y2 + -24y2 = 0 + -32x
Combine like terms: 32x + -32x = 0
0 + -12xy2 + 12x2 + -1x2y2 + -24y2 = 0 + -32x
-12xy2 + 12x2 + -1x2y2 + -24y2 = 0 + -32x
Remove the zero:
-12xy2 + 12x2 + -1x2y2 + -24y2 = -32x
Add '12xy2' to each side of the equation.
-12xy2 + 12x2 + -1x2y2 + 12xy2 + -24y2 = -32x + 12xy2
Reorder the terms:
-12xy2 + 12xy2 + 12x2 + -1x2y2 + -24y2 = -32x + 12xy2
Combine like terms: -12xy2 + 12xy2 = 0
0 + 12x2 + -1x2y2 + -24y2 = -32x + 12xy2
12x2 + -1x2y2 + -24y2 = -32x + 12xy2
Add '-12x2' to each side of the equation.
12x2 + -1x2y2 + -12x2 + -24y2 = -32x + 12xy2 + -12x2
Reorder the terms:
12x2 + -12x2 + -1x2y2 + -24y2 = -32x + 12xy2 + -12x2
Combine like terms: 12x2 + -12x2 = 0
0 + -1x2y2 + -24y2 = -32x + 12xy2 + -12x2
-1x2y2 + -24y2 = -32x + 12xy2 + -12x2
Add 'x2y2' to each side of the equation.
-1x2y2 + x2y2 + -24y2 = -32x + 12xy2 + -12x2 + x2y2
Combine like terms: -1x2y2 + x2y2 = 0
0 + -24y2 = -32x + 12xy2 + -12x2 + x2y2
-24y2 = -32x + 12xy2 + -12x2 + x2y2
Add '24y2' to each side of the equation.
-24y2 + 24y2 = -32x + 12xy2 + -12x2 + x2y2 + 24y2
Combine like terms: -24y2 + 24y2 = 0
0 = -32x + 12xy2 + -12x2 + x2y2 + 24y2
Simplifying
0 = -32x + 12xy2 + -12x2 + x2y2 + 24y2
The solution to this equation could not be determined.
This subproblem is being ignored because a solution could not be determined.
Solution
d = {0}