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The first order, first degree differential equation y’ = f(x,y) is said to be homogeneous, if
- a)f(x , y) is a homogeneous function of degree zero.
- b)f(x , y) is a homogeneous function of second degree.
- c)f(x , y) is a homogeneous function of first degree.
- d)f(x , y) is a homogeneous function of third degree.
Correct answer is option 'A'. Can you explain this answer?
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The first order, first degree differential equation y’ = f(x,y) ...
The correct answer is option A, because a first-order, first-degree differential equation is said to be homogeneous if the function f(x,y) is a homogeneous function of degree zero.
A function f(x,y) is said to be homogeneous of degree n if it satisfies the equation:
f(λx, λy) = λ^n * f(x,y)
where λ is a scalar. This means that if we multiply both the independent and dependent variables by a constant λ, the value of the function is scaled by a factor of λ^n.
In the case of a first-order, first-degree differential equation, the function f(x,y) is a function of two variables: x and y. For this type of equation, f(x,y) is said to be homogeneous if it is a homogeneous function of degree zero, which means that it satisfies the equation:
f(λx, λy) = f(x,y)
In other words, if we multiply both the independent and dependent variables by a constant λ, the value of the function does not change. This means that f(x,y) does not depend on the magnitude of the variables x and y, but only on their relative values.
For example, if f(x,y) = xy, then f(λx, λy) = λxy = xy, so f(x,y) is a homogeneous function of degree zero. On the other hand, if f(x,y) = x^2 + y^2, then f(λx, λy) = λ^2x^2 + λ^2y^2 = x^2 + y^2, so f(x,y) is not a homogeneous function.
Therefore, the correct answer is option A, because a first-order, first-degree differential equation is said to be homogeneous if the function f(x,y) is a homogeneous function of degree zero.
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The first order, first degree differential equation y’ = f(x,y) ...
Homogeneous First Order Differential Equations
Homogeneous first-order differential equation is an equation of the form:
y' = f(x,y)/g(x,y)
where f(x,y) and g(x,y) are homogeneous functions of the same degree. This type of equation is called homogeneous because if we replace y with ky, where k is a constant, the equation remains unchanged.
Homogeneous Function
A homogeneous function is a function f(x,y) that satisfies the following property:
f(tx,ty) = t^n f(x,y)
where n is a constant, and t is any real number. In other words, if we multiply both x and y by a constant t, the value of the function is multiplied by t^n.
Degree of Homogeneous Function
The degree of a homogeneous function is the power of x and y in its expression. For example, if f(x,y) = x^2y^3, then the degree of f is 5 (2+3).
Homogeneous First Order Differential Equation is Homogeneous Function of Degree Zero
The given first order, first-degree differential equation y = f(x,y) is said to be homogeneous if f(x,y) is a homogeneous function of degree zero. This means that if we replace both x and y with tx and ty, the value of f(x,y) remains unchanged.
In other words, the equation is homogeneous if it has the following property:
f(tx,ty) = f(x,y)
This property is satisfied only if the degree of f(x,y) is zero.
Conclusion
Hence, the correct answer to the given question is option A, which states that the given first order, first-degree differential equation is homogeneous if f(x,y) is a homogeneous function of degree zero.
Homogeneous first-order differential equation is an equation of the form:
y' = f(x,y)/g(x,y)
where f(x,y) and g(x,y) are homogeneous functions of the same degree. This type of equation is called homogeneous because if we replace y with ky, where k is a constant, the equation remains unchanged.
Homogeneous Function
A homogeneous function is a function f(x,y) that satisfies the following property:
f(tx,ty) = t^n f(x,y)
where n is a constant, and t is any real number. In other words, if we multiply both x and y by a constant t, the value of the function is multiplied by t^n.
Degree of Homogeneous Function
The degree of a homogeneous function is the power of x and y in its expression. For example, if f(x,y) = x^2y^3, then the degree of f is 5 (2+3).
Homogeneous First Order Differential Equation is Homogeneous Function of Degree Zero
The given first order, first-degree differential equation y = f(x,y) is said to be homogeneous if f(x,y) is a homogeneous function of degree zero. This means that if we replace both x and y with tx and ty, the value of f(x,y) remains unchanged.
In other words, the equation is homogeneous if it has the following property:
f(tx,ty) = f(x,y)
This property is satisfied only if the degree of f(x,y) is zero.
Conclusion
Hence, the correct answer to the given question is option A, which states that the given first order, first-degree differential equation is homogeneous if f(x,y) is a homogeneous function of degree zero.
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The first order, first degree differential equation y’ = f(x,y) is said to be homogeneous, ifa)f(x , y) is a homogeneous function of degree zero.b)f(x , y) is a homogeneous function of second degree.c)f(x , y) is a homogeneous function of first degree.d)f(x , y) is a homogeneous function of third degree.Correct answer is option 'A'. 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 The first order, first degree differential equation y’ = f(x,y) is said to be homogeneous, ifa)f(x , y) is a homogeneous function of degree zero.b)f(x , y) is a homogeneous function of second degree.c)f(x , y) is a homogeneous function of first degree.d)f(x , y) is a homogeneous function of third degree.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first order, first degree differential equation y’ = f(x,y) is said to be homogeneous, ifa)f(x , y) is a homogeneous function of degree zero.b)f(x , y) is a homogeneous function of second degree.c)f(x , y) is a homogeneous function of first degree.d)f(x , y) is a homogeneous function of third degree.Correct answer is option 'A'. Can you explain this answer?.
The first order, first degree differential equation y’ = f(x,y) is said to be homogeneous, ifa)f(x , y) is a homogeneous function of degree zero.b)f(x , y) is a homogeneous function of second degree.c)f(x , y) is a homogeneous function of first degree.d)f(x , y) is a homogeneous function of third degree.Correct answer is option 'A'. 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 The first order, first degree differential equation y’ = f(x,y) is said to be homogeneous, ifa)f(x , y) is a homogeneous function of degree zero.b)f(x , y) is a homogeneous function of second degree.c)f(x , y) is a homogeneous function of first degree.d)f(x , y) is a homogeneous function of third degree.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The first order, first degree differential equation y’ = f(x,y) is said to be homogeneous, ifa)f(x , y) is a homogeneous function of degree zero.b)f(x , y) is a homogeneous function of second degree.c)f(x , y) is a homogeneous function of first degree.d)f(x , y) is a homogeneous function of third degree.Correct answer is option 'A'. Can you explain this answer?.
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