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Any equation contains n-arbitrary constants, then the order of differential equation derived from it is
- a)n
- b)n - 1
- c)2
- d)n + 1
Correct answer is option 'A'. Can you explain this answer?
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Any equation contains n-arbitrary constants, then the order of differe...
The Order of a Differential Equation Derived from an Equation with n-Arbitrary Constants
When an equation contains n-arbitrary constants, it means that there are n unknown quantities that can take any value. In the context of a differential equation, these arbitrary constants arise from the integration process when solving the equation.
Definition: Order of a Differential Equation
The order of a differential equation is defined as the order of the highest derivative present in the equation. For example, if the equation contains the first derivative but not the second derivative, then the order of the differential equation is 1.
Explanation
Given that the equation contains n-arbitrary constants, we can infer that the equation is likely a solution to a differential equation. Let's denote the order of the differential equation as 'm'.
When we integrate a differential equation, we introduce an arbitrary constant for each integration. For instance, if we integrate once, we introduce one arbitrary constant. If we integrate twice, we introduce two arbitrary constants, and so on.
Hence, the number of arbitrary constants introduced during the integration process is directly related to the number of times we integrate the differential equation.
Relation between n and m
Let's consider the scenario where we have integrated the differential equation m times. At each integration, we introduce one arbitrary constant. Therefore, the number of arbitrary constants introduced is equal to the number of times we have integrated the differential equation.
Since the equation contains n-arbitrary constants, we can conclude that the number of times we have integrated the differential equation (m) is equal to the number of arbitrary constants (n).
Conclusion
Based on the above analysis, we can conclude that the order of the differential equation derived from an equation with n-arbitrary constants is n. Therefore, the correct answer is option 'A', n.
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