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The order of a differential equation representing a family of curves is same as:
- a)The number of arbitrary constants present in the equation.
- b)The number of variables in the equation
- c)The degree of the equation
- d)The number of curves in the family
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The order of a differential equation representing a family of curves i...
The Order of Differential Equations
In the context of differential equations, the order refers to the highest derivative present in the equation. When dealing with a family of curves, the order of the differential equation is closely linked to the number of arbitrary constants it contains.
Understanding Arbitrary Constants
- Arbitrary constants arise when integrating a function, representing a family of solutions or curves.
- For instance, if we have a first-order differential equation, it typically yields one arbitrary constant, leading to a family of curves parameterized by that constant.
Connection to Differential Equations
- The order of the differential equation mirrors the number of arbitrary constants.
- Thus, a second-order differential equation will introduce two arbitrary constants, allowing for a broader family of curves.
Other Options Explained
- Number of Variables: The number of variables in the equation does not directly correlate with the order. A single variable can lead to various orders based on derivatives.
- Degree of the Equation: The degree refers to the power of the highest derivative and is distinct from the order, which is purely about the highest derivative itself.
- Number of Curves in the Family: The number of curves is influenced by the number of arbitrary constants but is not the same as the order.
Conclusion
The correct answer, option 'A', is justified by the direct relationship between the order of a differential equation and the number of arbitrary constants present. This relationship is fundamental in understanding the behavior and characteristics of families of curves represented by differential equations.
In the context of differential equations, the order refers to the highest derivative present in the equation. When dealing with a family of curves, the order of the differential equation is closely linked to the number of arbitrary constants it contains.
Understanding Arbitrary Constants
- Arbitrary constants arise when integrating a function, representing a family of solutions or curves.
- For instance, if we have a first-order differential equation, it typically yields one arbitrary constant, leading to a family of curves parameterized by that constant.
Connection to Differential Equations
- The order of the differential equation mirrors the number of arbitrary constants.
- Thus, a second-order differential equation will introduce two arbitrary constants, allowing for a broader family of curves.
Other Options Explained
- Number of Variables: The number of variables in the equation does not directly correlate with the order. A single variable can lead to various orders based on derivatives.
- Degree of the Equation: The degree refers to the power of the highest derivative and is distinct from the order, which is purely about the highest derivative itself.
- Number of Curves in the Family: The number of curves is influenced by the number of arbitrary constants but is not the same as the order.
Conclusion
The correct answer, option 'A', is justified by the direct relationship between the order of a differential equation and the number of arbitrary constants present. This relationship is fundamental in understanding the behavior and characteristics of families of curves represented by differential equations.
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The order of a differential equation representing a family of curves i...
The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves.
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