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The differential equation (3x + 4y + 1)dx + (4x + 5y + 1)dy = 0 represents a family of
- a)circles
- b)parabolas
- c)ellipses
- d)hyperbolas
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The differential equation(3x + 4y + 1)dx + (4x + 5y + 1)dy = 0 represe...
The given differential equation is
(3x + 4y + 1)dx + (4x + 5y + 1)dy = 0 ...(i)
Comparing equation (i) with Mdx + Ndy = 0, we get M = 3x + 4y + 1 and N = 4x + 5y + 1.
Hence, equation (i) is an exact differential equation and solution is given by
On comparing equation (ii)ii with standard form of conic section
Hence, equation (i) is an exact differential equation and solution is given by
On comparing equation (ii)ii with standard form of conic section
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
we get,
a = 3, h = 4, b = 5
Here, h2 − ab = 16 − 15 = 1 > 0
Hence, the solution of differential equation represents family of hyperbolas.
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Community Answer
The differential equation(3x + 4y + 1)dx + (4x + 5y + 1)dy = 0 represe...
Given differential equation: (3x + 4y + 1)dx + (4x + 5y + 1)dy = 0
To determine the family of curves represented by this differential equation, we can try to simplify it and then analyze the resulting equation.
We can start by rearranging the equation to isolate dy/dx:
(4x + 5y + 1)dy = -(3x + 4y + 1)dx
dy/dx = -(3x + 4y + 1)/(4x + 5y + 1)
To further simplify the equation, let's define a new variable u = 3x + 4y + 1 and v = 4x + 5y + 1. Taking the derivatives with respect to x, we have du/dx = 3 + 4(dy/dx) and dv/dx = 4 + 5(dy/dx).
Substituting these values back into the equation dy/dx = -(3x + 4y + 1)/(4x + 5y + 1), we get:
du/dx - 3 = -u/v
du/v - dv/u = 3
This is a homogeneous differential equation, which can be solved by substituting u = v*y and v = u*x. Taking the derivatives, we have du = v*dy + y*dv and dv = u*dx + x*du.
Substituting these values back into the equation du/v - dv/u = 3, we get:
(v*dy + y*dv)/v - (u*dx + x*du)/u = 3
dy - dx = 3
Integrating both sides, we obtain y - x = 3x + C, where C is the constant of integration.
Simplifying this equation, we get y = 4x + C.
The resulting equation y = 4x + C represents a family of straight lines. Therefore, option D, hyperbolas, is incorrect.
The correct answer is option B, parabolas, as the given differential equation represents a family of parabolas.
To determine the family of curves represented by this differential equation, we can try to simplify it and then analyze the resulting equation.
We can start by rearranging the equation to isolate dy/dx:
(4x + 5y + 1)dy = -(3x + 4y + 1)dx
dy/dx = -(3x + 4y + 1)/(4x + 5y + 1)
To further simplify the equation, let's define a new variable u = 3x + 4y + 1 and v = 4x + 5y + 1. Taking the derivatives with respect to x, we have du/dx = 3 + 4(dy/dx) and dv/dx = 4 + 5(dy/dx).
Substituting these values back into the equation dy/dx = -(3x + 4y + 1)/(4x + 5y + 1), we get:
du/dx - 3 = -u/v
du/v - dv/u = 3
This is a homogeneous differential equation, which can be solved by substituting u = v*y and v = u*x. Taking the derivatives, we have du = v*dy + y*dv and dv = u*dx + x*du.
Substituting these values back into the equation du/v - dv/u = 3, we get:
(v*dy + y*dv)/v - (u*dx + x*du)/u = 3
dy - dx = 3
Integrating both sides, we obtain y - x = 3x + C, where C is the constant of integration.
Simplifying this equation, we get y = 4x + C.
The resulting equation y = 4x + C represents a family of straight lines. Therefore, option D, hyperbolas, is incorrect.
The correct answer is option B, parabolas, as the given differential equation represents a family of parabolas.
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The differential equation(3x + 4y + 1)dx + (4x + 5y + 1)dy = 0 represents a family ofa)circlesb)parabolasc)ellipsesd)hyperbolasCorrect answer is option 'D'. Can you explain this answer?
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