Question

Solve this differential equation - (2x + 3y +1) dx + (4x + 6y + 1) dy = 0, y(-2) = 2 . ​

Answer 1

Answer:

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Answer 2

Solution for (2x+3y+1)dx+(4x+6y+1)dy=0 equation:

Simplifying
(2x + 3y + 1) * dx + (4x + 6y + 1) * dy = 0

Reorder the terms:
(1 + 2x + 3y) * dx + (4x + 6y + 1) * dy = 0

Reorder the terms for easier multiplication:
dx(1 + 2x + 3y) + (4x + 6y + 1) * dy = 0
(1 * dx + 2x * dx + 3y * dx) + (4x + 6y + 1) * dy = 0

Reorder the terms:
(1dx + 3dxy + 2dx2) + (4x + 6y + 1) * dy = 0
(1dx + 3dxy + 2dx2) + (4x + 6y + 1) * dy = 0

Reorder the terms:
1dx + 3dxy + 2dx2 + (1 + 4x + 6y) * dy = 0

Reorder the terms for easier multiplication:
1dx + 3dxy + 2dx2 + dy(1 + 4x + 6y) = 0
1dx + 3dxy + 2dx2 + (1 * dy + 4x * dy + 6y * dy) = 0

Reorder the terms:
1dx + 3dxy + 2dx2 + (4dxy + 1dy + 6dy2) = 0
1dx + 3dxy + 2dx2 + (4dxy + 1dy + 6dy2) = 0

Reorder the terms:
1dx + 3dxy + 4dxy + 2dx2 + 1dy + 6dy2 = 0

Combine like terms: 3dxy + 4dxy = 7dxy
1dx + 7dxy + 2dx2 + 1dy + 6dy2 = 0

Solving
1dx + 7dxy + 2dx2 + 1dy + 6dy2 = 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(x + 7xy + 2x2 + y + 6y2) = 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 '(x + 7xy + 2x2 + y + 6y2)' equal to zero and attempt to solve:

Simplifying
x + 7xy + 2x2 + y + 6y2 = 0

Solving
x + 7xy + 2x2 + y + 6y2 = 0

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

Add '-1x' to each side of the equation.
x + 7xy + 2x2 + y + -1x + 6y2 = 0 + -1x

Reorder the terms:
x + -1x + 7xy + 2x2 + y + 6y2 = 0 + -1x

Combine like terms: x + -1x = 0
0 + 7xy + 2x2 + y + 6y2 = 0 + -1x
7xy + 2x2 + y + 6y2 = 0 + -1x
Remove the zero:
7xy + 2x2 + y + 6y2 = -1x

Add '-7xy' to each side of the equation.
7xy + 2x2 + y + -7xy + 6y2 = -1x + -7xy

Reorder the terms:
7xy + -7xy + 2x2 + y + 6y2 = -1x + -7xy

Combine like terms: 7xy + -7xy = 0
0 + 2x2 + y + 6y2 = -1x + -7xy
2x2 + y + 6y2 = -1x + -7xy

Add '-2x2' to each side of the equation.
2x2 + y + -2x2 + 6y2 = -1x + -7xy + -2x2

Reorder the terms:
2x2 + -2x2 + y + 6y2 = -1x + -7xy + -2x2

Combine like terms: 2x2 + -2x2 = 0
0 + y + 6y2 = -1x + -7xy + -2x2
y + 6y2 = -1x + -7xy + -2x2

Add '-1y' to each side of the equation.
y + -1y + 6y2 = -1x + -7xy + -2x2 + -1y

Combine like terms: y + -1y = 0
0 + 6y2 = -1x + -7xy + -2x2 + -1y
6y2 = -1x + -7xy + -2x2 + -1y

Add '-6y2' to each side of the equation.
6y2 + -6y2 = -1x + -7xy + -2x2 + -1y + -6y2

Combine like terms: 6y2 + -6y2 = 0
0 = -1x + -7xy + -2x2 + -1y + -6y2

Simplifying
0 = -1x + -7xy + -2x2 + -1y + -6y2

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
d = {0}