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The differential equation y dy/dx+x=c represent
- a)a family of hyperbolas
- b)a family of circles whose centres on the y-axis
- c)a family of parabolas
- d)a family of circles whose centres are on the x-axis
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The differential equation y dy/dx+x=c representa)a family of hyperbola...
ydy = (c−x)dx
Integrating both sides,
∫ydy = ∫(c−x)dx
y2/2 = cx − x2/2 + k
x2/2 + y2/2 − cx = k
x2 + y2 − 2cx = 2k
(x−c)2 + y2 = 2k + c2
This is equation of a circle with center on x-axis.
Integrating both sides,
∫ydy = ∫(c−x)dx
y2/2 = cx − x2/2 + k
x2/2 + y2/2 − cx = k
x2 + y2 − 2cx = 2k
(x−c)2 + y2 = 2k + c2
This is equation of a circle with center on x-axis.
Most Upvoted Answer
The differential equation y dy/dx+x=c representa)a family of hyperbola...
Explanation:
The given differential equation is y(dy/dx) = c/x.
To find the nature of the differential equation, we need to eliminate the arbitrary constant c by differentiating the given equation.
Differentiating both sides with respect to x, we get:
(dy/dx) + y(d^2y/dx^2) = -c/x^2
Multiplying both sides by dx, we get:
dy + y(d^2y/dx^2) dx = -c/x^2 dx
Integrating both sides with respect to x, we get:
y^2/2 + (dy/dx)^2/2 = -c ln|x| + k
where k is the arbitrary constant of integration.
This is the required family of differential equations.
Nature of the Differential Equation:
From the given equation, we can see that the coefficient of (dy/dx) is y, which is a function of x.
Hence, this is a variable coefficient differential equation.
To determine the nature of the solution, we need to analyze the given equation.
The equation y(dy/dx) = c/x represents a family of circles whose centers are on the x-axis.
Hence, the correct answer is option D.
The given differential equation is y(dy/dx) = c/x.
To find the nature of the differential equation, we need to eliminate the arbitrary constant c by differentiating the given equation.
Differentiating both sides with respect to x, we get:
(dy/dx) + y(d^2y/dx^2) = -c/x^2
Multiplying both sides by dx, we get:
dy + y(d^2y/dx^2) dx = -c/x^2 dx
Integrating both sides with respect to x, we get:
y^2/2 + (dy/dx)^2/2 = -c ln|x| + k
where k is the arbitrary constant of integration.
This is the required family of differential equations.
Nature of the Differential Equation:
From the given equation, we can see that the coefficient of (dy/dx) is y, which is a function of x.
Hence, this is a variable coefficient differential equation.
To determine the nature of the solution, we need to analyze the given equation.
The equation y(dy/dx) = c/x represents a family of circles whose centers are on the x-axis.
Hence, the correct answer is option D.
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The differential equation y dy/dx+x=c representa)a family of hyperbolasb)a family of circles whose centres on the y-axisc)a family of parabolasd)a family of circles whose centres are on the x-axisCorrect answer is option 'D'. Can you explain this answer?
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