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The differential equation 2ydx – (3y – 2x)dy = 0 is
- a)exact and homogeneous but not linear
- b)homogeneous and linear but not exact
- c)exact and linear but not homogeneous
- d)exact, homogeneous and linear
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The differential equation 2ydx – (3y – 2x)dy = 0 isa)exact...
2ydx + (2x – 3y)dy = 0 ...(1)
It is Homogeneous
M = 2y
Most Upvoted Answer
The differential equation 2ydx – (3y – 2x)dy = 0 isa)exact...
Given Differential Equation:
The given differential equation is 2ydx + (3y - 2x)dy = 0.
Explanation:
To analyze the given differential equation, we need to check for certain properties such as linearity, homogeneity, and exactness.
Linearity:
A differential equation is said to be linear if the dependent variable and its derivatives occur only in the first degree and are not multiplied together.
In the given equation, the term (3y - 2x)dy has the dependent variable y and its derivative dy occurring in the first degree, and they are not multiplied together. Hence, the given equation is linear.
Homogeneity:
A differential equation is said to be homogeneous if all the terms in the equation have the same degree with respect to the dependent variable and its derivatives.
In the given equation, the terms 2ydx and (3y - 2x)dy have the same degree with respect to y and its derivative dy. Hence, the given equation is homogeneous.
Exactness:
A differential equation is said to be exact if it can be expressed as the derivative of some function.
To check for exactness, we need to calculate the partial derivatives of the terms with respect to x and y.
The partial derivative of 2ydx with respect to y is 0, and the partial derivative of (3y - 2x)dy with respect to x is -2.
Since the partial derivatives are not equal, the given equation is not exact.
Conclusion:
Based on the above analysis, we can conclude that the given differential equation is homogeneous and linear but not exact (option B).
The given differential equation is 2ydx + (3y - 2x)dy = 0.
Explanation:
To analyze the given differential equation, we need to check for certain properties such as linearity, homogeneity, and exactness.
Linearity:
A differential equation is said to be linear if the dependent variable and its derivatives occur only in the first degree and are not multiplied together.
In the given equation, the term (3y - 2x)dy has the dependent variable y and its derivative dy occurring in the first degree, and they are not multiplied together. Hence, the given equation is linear.
Homogeneity:
A differential equation is said to be homogeneous if all the terms in the equation have the same degree with respect to the dependent variable and its derivatives.
In the given equation, the terms 2ydx and (3y - 2x)dy have the same degree with respect to y and its derivative dy. Hence, the given equation is homogeneous.
Exactness:
A differential equation is said to be exact if it can be expressed as the derivative of some function.
To check for exactness, we need to calculate the partial derivatives of the terms with respect to x and y.
The partial derivative of 2ydx with respect to y is 0, and the partial derivative of (3y - 2x)dy with respect to x is -2.
Since the partial derivatives are not equal, the given equation is not exact.
Conclusion:
Based on the above analysis, we can conclude that the given differential equation is homogeneous and linear but not exact (option B).
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The differential equation 2ydx – (3y – 2x)dy = 0 isa)exact...
Exact and homogeneous but not linear
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