Identify the highest order derivative: The highest order derivative in the given differential equation \((y^{\prime\prime})^{2}+(y^{\prime})^{3}=x~\sin(y^{\prime})\) is \(y^{\prime\prime}\), which is the second derivative.
Check for transcendental functions of derivatives: The term \(\sin(y^{\prime})\) involves a transcendental function (sine) of a derivative (\(y^{\prime}\)). When a differential equation contains transcendental functions of derivatives, the degree of the differential equation is not defined.
Conclusion: Since the differential equation contains \(\sin(y^{\prime})\), the degree is not defined.
Correct Answer: not defined
AI Suggestion: Option D
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because the student needs to apply the concept of order and degree of differential equations to determine the degree of the given equation. This involves identifying the highest order derivative and its power, considering any restrictions due to transcendental functions of derivatives.
Knowledge Dimension:CONCEPTUAL
Justification:The question requires understanding the concept of the degree of a differential equation and how it is affected by transcendental functions involving derivatives. It's not just recalling a definition but applying the understanding of the concept.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the knowledge of defining the degree of a differential equation, a concept covered in the textbook.