solve (4x+3y+1) dx+3x+2y+1) dy=0
Answers
Answer:
To Solve the differential equation
(4x+3y +1)dx +(3x +2y+1)dy = 0
and to show that it represents a family of hyperbolas having as asymptotes the lines x+y=0 , 2x+y+1=0.
EVALUATION
Here the given differential equation is
(4x+3y +1)dx +(3x +2y+1)dy = 0
\sf{ \implies \: 4x \: dx + 3(y \: dx + x \: dy) + 2y \: dy + dx + dy = 0}⟹4xdx+3(ydx+xdy)+2ydy+dx+dy=0
\sf{ \implies \: 4x \: dx + 3d(xy ) + 2y \: dy + dx + dy = 0}⟹4xdx+3d(xy)+2ydy+dx+dy=0
On integration we get
\displaystyle\sf{ \implies \int 4x \: dx + 3 \int d(xy ) + 2 \int y \: dy + \int \: dx + \int \: dy = 0}⟹∫4xdx+3∫d(xy)+2∫ydy+∫dx+∫dy=0
\displaystyle\sf{ \implies 2 {x}^{2} + 3 xy + {y}^{2} + x + y= c}⟹2x
2
+3xy+y
2
+x+y=c
Where C is integration constant
The above equation represents the required solution
On simplification we get
\displaystyle\sf{ \implies (x + y)(2x + y + 1)= c}⟹(x+y)(2x+y+1)=c
From which it is clear that it represents a family of hyperbolas having as asymptotes the lines x+y=0 , 2x+y+1=0.
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Answer:
0
Explanation:
Simplifying
(4x + 3y + 1) * dx + (3x + 2y + 1) * dy = 0
Reorder the terms:
(1 + 4x + 3y) * dx + (3x + 2y + 1) * dy = 0
Reorder the terms for easier multiplication:
dx(1 + 4x + 3y) + (3x + 2y + 1) * dy = 0
(1 * dx + 4x * dx + 3y * dx) + (3x + 2y + 1) * dy = 0
Reorder the terms:
(1dx + 3dxy + 4dx2) + (3x + 2y + 1) * dy = 0
(1dx + 3dxy + 4dx2) + (3x + 2y + 1) * dy = 0
Reorder the terms:
1dx + 3dxy + 4dx2 + (1 + 3x + 2y) * dy = 0
Reorder the terms for easier multiplication:
1dx + 3dxy + 4dx2 + dy(1 + 3x + 2y) = 0
1dx + 3dxy + 4dx2 + (1 * dy + 3x * dy + 2y * dy) = 0
Reorder the terms:
1dx + 3dxy + 4dx2 + (3dxy + 1dy + 2dy2) = 0
1dx + 3dxy + 4dx2 + (3dxy + 1dy + 2dy2) = 0
Reorder the terms:
1dx + 3dxy + 3dxy + 4dx2 + 1dy + 2dy2 = 0
Combine like terms: 3dxy + 3dxy = 6dxy
1dx + 6dxy + 4dx2 + 1dy + 2dy2 = 0
Solving
1dx + 6dxy + 4dx2 + 1dy + 2dy2 = 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 + 6xy + 4x2 + y + 2y2) = 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 + 6xy + 4x2 + y + 2y2)' equal to zero and attempt to solve:
Simplifying
x + 6xy + 4x2 + y + 2y2 = 0
Solving
x + 6xy + 4x2 + y + 2y2 = 0
Move all terms containing d to the left, all other terms to the right.
Add '-1x' to each side of the equation.
x + 6xy + 4x2 + y + -1x + 2y2 = 0 + -1x
Reorder the terms:
x + -1x + 6xy + 4x2 + y + 2y2 = 0 + -1x
Combine like terms: x + -1x = 0
0 + 6xy + 4x2 + y + 2y2 = 0 + -1x
6xy + 4x2 + y + 2y2 = 0 + -1x
Remove the zero:
6xy + 4x2 + y + 2y2 = -1x
Add '-6xy' to each side of the equation.
6xy + 4x2 + y + -6xy + 2y2 = -1x + -6xy
Reorder the terms:
6xy + -6xy + 4x2 + y + 2y2 = -1x + -6xy
Combine like terms: 6xy + -6xy = 0
0 + 4x2 + y + 2y2 = -1x + -6xy
4x2 + y + 2y2 = -1x + -6xy
Add '-4x2' to each side of the equation.
4x2 + y + -4x2 + 2y2 = -1x + -6xy + -4x2
Reorder the terms:
4x2 + -4x2 + y + 2y2 = -1x + -6xy + -4x2
Combine like terms: 4x2 + -4x2 = 0
0 + y + 2y2 = -1x + -6xy + -4x2
y + 2y2 = -1x + -6xy + -4x2
Add '-1y' to each side of the equation.
y + -1y + 2y2 = -1x + -6xy + -4x2 + -1y
Combine like terms: y + -1y = 0
0 + 2y2 = -1x + -6xy + -4x2 + -1y
2y2 = -1x + -6xy + -4x2 + -1y
Add '-2y2' to each side of the equation.
2y2 + -2y2 = -1x + -6xy + -4x2 + -1y + -2y2
Combine like terms: 2y2 + -2y2 = 0
0 = -1x + -6xy + -4x2 + -1y + -2y2
Simplifying
0 = -1x + -6xy + -4x2 + -1y + -2y2
The solution to this equation could not be determined.
This subproblem is being ignored because a solution could not be determined.
Solution
d = {0}