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A differential equation of the form y' = F(x,y) is homogeneous if
- a)F(x,y) is a homogeneous function of degree one
- b)F(x,y) is a homogeneous function of degree three
- c)F(x,y) is a homogeneous function of degree two
- d)F(x,y) is a homogeneous function of degree zero
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
A differential equation of the form y' = F(x,y) is homogeneous ifa...
A differential equation of the form y' = F(x,y) is homogeneous if F(x,y) is a homogeneous function of degree zero.
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A differential equation of the form y' = F(x,y) is homogeneous ifa...
Understanding Homogeneous Differential Equations
Homogeneous differential equations have specific characteristics that define them. The equation y = F(x, y) is considered homogeneous if the function F(x, y) exhibits a particular type of behavior concerning scaling.
Definition of Homogeneous Functions
A function F(x, y) is termed homogeneous of degree n if, for any scalar t:
- F(tx, ty) = t^n * F(x, y)
This means that if we scale both variables x and y by the same factor t, the function F is scaled by t raised to the power of n.
Homogeneity in Differential Equations
For the differential equation y = F(x, y) to be classified as homogeneous, the function F(x, y) must satisfy the following condition:
- F(x, y) is a homogeneous function of degree zero.
This implies:
- F(tx, ty) = F(x, y) for any scalar t.
Why Degree Zero is Key
- Degree Zero Function: When F(x, y) is homogeneous of degree zero, it signifies that the function does not change in value when both x and y are multiplied by the same non-zero scalar. This reflects the property that the relationship between x and y remains invariant under scaling.
- Implication: This invariance is crucial for the analysis of solutions to the differential equation, helping to identify unique solutions based on initial conditions.
Conclusion
In summary, for the differential equation y = F(x, y) to be homogeneous, F(x, y) must specifically be a homogeneous function of degree zero. This characteristic is essential in the context of solving differential equations and understanding their behavior under transformations.
Homogeneous differential equations have specific characteristics that define them. The equation y = F(x, y) is considered homogeneous if the function F(x, y) exhibits a particular type of behavior concerning scaling.
Definition of Homogeneous Functions
A function F(x, y) is termed homogeneous of degree n if, for any scalar t:
- F(tx, ty) = t^n * F(x, y)
This means that if we scale both variables x and y by the same factor t, the function F is scaled by t raised to the power of n.
Homogeneity in Differential Equations
For the differential equation y = F(x, y) to be classified as homogeneous, the function F(x, y) must satisfy the following condition:
- F(x, y) is a homogeneous function of degree zero.
This implies:
- F(tx, ty) = F(x, y) for any scalar t.
Why Degree Zero is Key
- Degree Zero Function: When F(x, y) is homogeneous of degree zero, it signifies that the function does not change in value when both x and y are multiplied by the same non-zero scalar. This reflects the property that the relationship between x and y remains invariant under scaling.
- Implication: This invariance is crucial for the analysis of solutions to the differential equation, helping to identify unique solutions based on initial conditions.
Conclusion
In summary, for the differential equation y = F(x, y) to be homogeneous, F(x, y) must specifically be a homogeneous function of degree zero. This characteristic is essential in the context of solving differential equations and understanding their behavior under transformations.
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A differential equation of the form y' = F(x,y) is homogeneous ifa)F(x,y) is a homogeneous function of degree oneb)F(x,y) is a homogeneous function of degree threec)F(x,y) is a homogeneous function of degree twod)F(x,y) is a homogeneous function of degree zeroCorrect answer is option 'D'. Can you explain this answer?
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A differential equation of the form y' = F(x,y) is homogeneous ifa)F(x,y) is a homogeneous function of degree oneb)F(x,y) is a homogeneous function of degree threec)F(x,y) is a homogeneous function of degree twod)F(x,y) is a homogeneous function of degree zeroCorrect answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A differential equation of the form y' = F(x,y) is homogeneous ifa)F(x,y) is a homogeneous function of degree oneb)F(x,y) is a homogeneous function of degree threec)F(x,y) is a homogeneous function of degree twod)F(x,y) is a homogeneous function of degree zeroCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A differential equation of the form y' = F(x,y) is homogeneous ifa)F(x,y) is a homogeneous function of degree oneb)F(x,y) is a homogeneous function of degree threec)F(x,y) is a homogeneous function of degree twod)F(x,y) is a homogeneous function of degree zeroCorrect answer is option 'D'. Can you explain this answer?.
A differential equation of the form y' = F(x,y) is homogeneous ifa)F(x,y) is a homogeneous function of degree oneb)F(x,y) is a homogeneous function of degree threec)F(x,y) is a homogeneous function of degree twod)F(x,y) is a homogeneous function of degree zeroCorrect answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A differential equation of the form y' = F(x,y) is homogeneous ifa)F(x,y) is a homogeneous function of degree oneb)F(x,y) is a homogeneous function of degree threec)F(x,y) is a homogeneous function of degree twod)F(x,y) is a homogeneous function of degree zeroCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A differential equation of the form y' = F(x,y) is homogeneous ifa)F(x,y) is a homogeneous function of degree oneb)F(x,y) is a homogeneous function of degree threec)F(x,y) is a homogeneous function of degree twod)F(x,y) is a homogeneous function of degree zeroCorrect answer is option 'D'. Can you explain this answer?.
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