Q656: The order and degree of the differential equation \((\frac{d...

KNOWLEDGE BASED

APPLY

1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE

The order and degree of the differential equation $(\frac{d^{2}y}{dx^{2}})^{2}+(\frac{dy}{dx})^{2}=x\sin(\frac{dy}{dx})$ are:

(A) order 2, degree 2

(B) order 2, degree 1

(D) order 1, degree not defined

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Correct Answer: C

Explanation

The order of a differential equation is the order of the highest derivative present in the equation.The degree of a differential equation is the power of the highest order derivative, provided the differential equation is a polynomial equation in its derivatives.

For the degree to be defined, the equation must be free from terms like $\sin(\frac{dy}{dx})$, $e^{\frac{dy}{dx}}$, $\log(\frac{d^2y}{dx^2})$, etc. (i.e., the derivatives must not appear inside transcendental functions).In the given equation, the term $x\sin\left(\frac{dy}{dx}\right)$ involves the first derivative $\frac{dy}{dx}$ inside the transcendental function $\sin()$.Since the equation is not a polynomial in its derivatives, the Degree is not defined

AI Tutor Explanation

Step-by-Step Solution

Identify the order: The order of a differential equation is the highest order derivative present in the equation. In this case, the highest order derivative is $\frac{d^{2}y}{dx^{2}}$, which is the second derivative. Therefore, the order is 2.
Identify the degree: The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial equation in derivatives. Here, we have $x\sin(\frac{dy}{dx})$. Since the derivative $\frac{dy}{dx}$ is inside the sine function, the equation is not a polynomial equation in derivatives. Therefore, the degree is not defined.
Conclusion: The order is 2, and the degree is not defined.

Correct Answer: order 2, degree not defined

AI Suggestion: Option C

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student needs to apply their understanding of order and degree of differential equations to a given equation to determine the correct answer.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concepts of 'order' and 'degree' of a differential equation, rather than recalling specific facts or following a fixed procedure. The presence of $\sin(\frac{dy}{dx})$ makes it a conceptual problem.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the student's understanding of a definition from the textbook.

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