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The differential equation for the family of curves x2+ y2 - 2ay = 0, where a is an arbitrary constant is
- a)(x2 - y2)y′ = 2xy
- b)2(x2 + y2)y′ = xy
- c)2(x2 - y2)y′ = xy
- d)(x2 + y2)y′ = 2xy
Correct answer is option 'A'. Can you explain this answer?
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The differential equation for the family of curves x2+ y2 - 2ay = 0, w...
B) 2xy - 2a
c) y(x2 - y2) - 2a
d) y(x2 + y2) - 2a
We can start by implicitly differentiating the given equation with respect to x:
2x dx + 2y dy - 2a dy/dx = 0
Simplifying and solving for dy/dx, we get:
dy/dx = (y - x^2)/y
Substituting y = x^2/(2a) into the above expression, we get:
dy/dx = (x^2/(2a) - x^2)/ (x^2/(2a)) = 2a/x^2 - 2
Multiplying both sides by x^2, we get:
x^2 dy/dx = 2a - 2x^2
Therefore, the differential equation for the family of curves x^2 y^2 - 2ay = 0 is:
x^2 dy/dx = 2a - 2x^2
which can be simplified to:
dy/dx = (2a/x^2) - 2
c) y(x2 - y2) - 2a
d) y(x2 + y2) - 2a
We can start by implicitly differentiating the given equation with respect to x:
2x dx + 2y dy - 2a dy/dx = 0
Simplifying and solving for dy/dx, we get:
dy/dx = (y - x^2)/y
Substituting y = x^2/(2a) into the above expression, we get:
dy/dx = (x^2/(2a) - x^2)/ (x^2/(2a)) = 2a/x^2 - 2
Multiplying both sides by x^2, we get:
x^2 dy/dx = 2a - 2x^2
Therefore, the differential equation for the family of curves x^2 y^2 - 2ay = 0 is:
x^2 dy/dx = 2a - 2x^2
which can be simplified to:
dy/dx = (2a/x^2) - 2
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The differential equation for the family of curves x2+ y2 - 2ay = 0, w...
Explanation:
Given Differential Equation: x2 + y2 - 2ay = 0
Step 1: Differentiate the given equation with respect to x
- Differentiating both sides of the equation with respect to x, we get:
d/dx(x2) + d/dx(y2) - d/dx(2ay) = 0
- 2x + 2y * dy/dx - 2a * dy/dx = 0
- 2x + 2y * dy/dx = 2a * dy/dx
Step 2: Simplify the equation
- 2x + 2y * dy/dx = 2a * dy/dx
- 2x = 2a * dy/dx - 2y * dy/dx
- 2x = 2(dy/dx)(a - y)
- x = (dy/dx)(a - y)
Step 3: Rearrange the equation
- Rearranging the terms, we get:
- (x2 - y2) * dy/dx = 2xy
Therefore, the correct answer is A: (x2 - y2) * y' = 2xy
Given Differential Equation: x2 + y2 - 2ay = 0
Step 1: Differentiate the given equation with respect to x
- Differentiating both sides of the equation with respect to x, we get:
d/dx(x2) + d/dx(y2) - d/dx(2ay) = 0
- 2x + 2y * dy/dx - 2a * dy/dx = 0
- 2x + 2y * dy/dx = 2a * dy/dx
Step 2: Simplify the equation
- 2x + 2y * dy/dx = 2a * dy/dx
- 2x = 2a * dy/dx - 2y * dy/dx
- 2x = 2(dy/dx)(a - y)
- x = (dy/dx)(a - y)
Step 3: Rearrange the equation
- Rearranging the terms, we get:
- (x2 - y2) * dy/dx = 2xy
Therefore, the correct answer is A: (x2 - y2) * y' = 2xy
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The differential equation for the family of curves x2+ y2 - 2ay = 0, where a is an arbitrary constant isa)(x2 - y2)y′ = 2xyb)2(x2 + y2)y′ = xyc)2(x2 - y2)y′ = xyd)(x2 + y2)y′ = 2xyCorrect answer is option 'A'. Can you explain this answer?
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The differential equation for the family of curves x2+ y2 - 2ay = 0, where a is an arbitrary constant isa)(x2 - y2)y′ = 2xyb)2(x2 + y2)y′ = xyc)2(x2 - y2)y′ = xyd)(x2 + y2)y′ = 2xyCorrect answer is option 'A'. Can you explain this answer? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The differential equation for the family of curves x2+ y2 - 2ay = 0, where a is an arbitrary constant isa)(x2 - y2)y′ = 2xyb)2(x2 + y2)y′ = xyc)2(x2 - y2)y′ = xyd)(x2 + y2)y′ = 2xyCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The differential equation for the family of curves x2+ y2 - 2ay = 0, where a is an arbitrary constant isa)(x2 - y2)y′ = 2xyb)2(x2 + y2)y′ = xyc)2(x2 - y2)y′ = xyd)(x2 + y2)y′ = 2xyCorrect answer is option 'A'. Can you explain this answer?.
The differential equation for the family of curves x2+ y2 - 2ay = 0, where a is an arbitrary constant isa)(x2 - y2)y′ = 2xyb)2(x2 + y2)y′ = xyc)2(x2 - y2)y′ = xyd)(x2 + y2)y′ = 2xyCorrect answer is option 'A'. Can you explain this answer? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The differential equation for the family of curves x2+ y2 - 2ay = 0, where a is an arbitrary constant isa)(x2 - y2)y′ = 2xyb)2(x2 + y2)y′ = xyc)2(x2 - y2)y′ = xyd)(x2 + y2)y′ = 2xyCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The differential equation for the family of curves x2+ y2 - 2ay = 0, where a is an arbitrary constant isa)(x2 - y2)y′ = 2xyb)2(x2 + y2)y′ = xyc)2(x2 - y2)y′ = xyd)(x2 + y2)y′ = 2xyCorrect answer is option 'A'. Can you explain this answer?.
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