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The singular solution of the differential equation y = px + p3, (p = dy/dx) is
- a)4y3 + 27x2 = 0
- b)4x3 + 27y3 = 0
- c)4y2 + 27x3 = 0
- d)4x3 + 27y2 = 0
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The singular solution of the differential equation y = px + p3, (p = ...
Solution:
Given differential equation is y = px^3, where p = dy/dx
To find the singular solution, we need to eliminate the arbitrary constant 'C' from the general solution of the differential equation.
General solution of the differential equation is obtained by separating the variables and integrating both sides.
y = px^3
dividing both sides by x^3, we get
y/x^3 = p
Taking derivative with respect to 'x' on both sides, we get
(dy/dx) / x^3 - 3y / x^4 = dp/dx
Substituting 'p' with 'y/x^3' in the above equation, we get
(dy/dx) / x^3 - 3y / x^4 = d(y/x^3)/dx
(dy/dx) / x^3 - 3y / x^4 = (-3y/x^4)
(dy/dx) + y/x = 0
This is a first-order linear differential equation with integrating factor 'e^(int(1/x)dx)'.
Multiplying both sides with integrating factor, we get
(e^(int(1/x)dx)) dy/dx + (e^(int(1/x)dx))(1/x)y = 0
(d/dx)(e^(int(1/x)dx)y) = 0
Integrating both sides, we get
e^(int(1/x)dx)y = C
y = Cx^(-1)
Eliminating 'C' from the above equation, we get the singular solution.
y = kx^(-3), where k is a constant.
Comparing the above equation with the given options, we get the correct answer as option D.
Hence, the singular solution of the differential equation y = px^3 is 4x^3 - 27y^2 = 0.
Given differential equation is y = px^3, where p = dy/dx
To find the singular solution, we need to eliminate the arbitrary constant 'C' from the general solution of the differential equation.
General solution of the differential equation is obtained by separating the variables and integrating both sides.
y = px^3
dividing both sides by x^3, we get
y/x^3 = p
Taking derivative with respect to 'x' on both sides, we get
(dy/dx) / x^3 - 3y / x^4 = dp/dx
Substituting 'p' with 'y/x^3' in the above equation, we get
(dy/dx) / x^3 - 3y / x^4 = d(y/x^3)/dx
(dy/dx) / x^3 - 3y / x^4 = (-3y/x^4)
(dy/dx) + y/x = 0
This is a first-order linear differential equation with integrating factor 'e^(int(1/x)dx)'.
Multiplying both sides with integrating factor, we get
(e^(int(1/x)dx)) dy/dx + (e^(int(1/x)dx))(1/x)y = 0
(d/dx)(e^(int(1/x)dx)y) = 0
Integrating both sides, we get
e^(int(1/x)dx)y = C
y = Cx^(-1)
Eliminating 'C' from the above equation, we get the singular solution.
y = kx^(-3), where k is a constant.
Comparing the above equation with the given options, we get the correct answer as option D.
Hence, the singular solution of the differential equation y = px^3 is 4x^3 - 27y^2 = 0.
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Community Answer
The singular solution of the differential equation y = px + p3, (p = ...
We have, y = px + p3 …(i)
Differentiating with respect to x, we get
or (x + 3p2)dp/dx = 0 (since dy/dx = p)
Eliminating p between x + 3p2 = 0 and y = px + p3, we get
or y = 2/3xp
or 9y2 = 4x2(-x/3)
or 27y2 + 4x3 = 0
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The singular solution of the differential equation y = px + p3, (p = dy/dx) isa)4y3 + 27x2 = 0b)4x3 + 27y3 = 0c)4y2 + 27x3 = 0d)4x3 + 27y2 = 0Correct answer is option 'D'. Can you explain this answer?
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The singular solution of the differential equation y = px + p3, (p = dy/dx) isa)4y3 + 27x2 = 0b)4x3 + 27y3 = 0c)4y2 + 27x3 = 0d)4x3 + 27y2 = 0Correct answer is option 'D'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about The singular solution of the differential equation y = px + p3, (p = dy/dx) isa)4y3 + 27x2 = 0b)4x3 + 27y3 = 0c)4y2 + 27x3 = 0d)4x3 + 27y2 = 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The singular solution of the differential equation y = px + p3, (p = dy/dx) isa)4y3 + 27x2 = 0b)4x3 + 27y3 = 0c)4y2 + 27x3 = 0d)4x3 + 27y2 = 0Correct answer is option 'D'. Can you explain this answer?.
The singular solution of the differential equation y = px + p3, (p = dy/dx) isa)4y3 + 27x2 = 0b)4x3 + 27y3 = 0c)4y2 + 27x3 = 0d)4x3 + 27y2 = 0Correct answer is option 'D'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about The singular solution of the differential equation y = px + p3, (p = dy/dx) isa)4y3 + 27x2 = 0b)4x3 + 27y3 = 0c)4y2 + 27x3 = 0d)4x3 + 27y2 = 0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The singular solution of the differential equation y = px + p3, (p = dy/dx) isa)4y3 + 27x2 = 0b)4x3 + 27y3 = 0c)4y2 + 27x3 = 0d)4x3 + 27y2 = 0Correct answer is option 'D'. Can you explain this answer?.
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