Q657: If $p$ and $q$ are respectively the order and degree of ...

KNOWLEDGE BASED

UNDERSTAND

1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE

If $p$ and $q$ are respectively the order and degree of the differential equation $\frac{d}{dx}(\frac{dy}{dx})^{3}=0,$ then $(p-q)$ is

(A) 0

(B) 1

(D) 3

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Correct Answer: B

Explanation

The given differential equation is:

$\displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)^3 = 0$

Step 1: Expand the derivative

$\displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)^3 = 3\left(\frac{dy}{dx}\right)^2 \frac{d^2y}{dx^2} = 0$

Step 2: Identify order and degree

The highest order derivative is $\dfrac{d^2y}{dx^2}$, so the order is $p = 2$.

The equation is a polynomial in derivatives, and the degree of the highest derivative is $1$, so $q = 1$.

Step 3: Compute $(p - q)$

$(p - q) = 2 - 1 = 1$

∴ Final Answer: $(p - q) = 1$

AI Tutor Explanation

Step-by-Step Solution

The given differential equation is $\frac{d}{dx}(\frac{dy}{dx})^{3}=0$.

First, we simplify the equation:

$\frac{d}{dx}(\frac{dy}{dx})^{3} = \frac{d}{dx}((\frac{dy}{dx})^{3}) = 0$

Differentiating $\left(\frac{dy}{dx}\right)^3$ with respect to $x$ gives:

$3\left(\frac{dy}{dx}\right)^2 \cdot \frac{d^2y}{dx^2} = 0$

The order of a differential equation is the highest order derivative present in the equation. Here, the highest order derivative is $\frac{d^2y}{dx^2}$, which is the second derivative. So, the order $p = 2$.

The degree of a differential equation is the power of the highest order derivative, when the differential equation is expressed in a form where all derivatives are free from radicals and fractions. In this case, the power of $\frac{d^2y}{dx^2}$ is 1. So, the degree $q = 1$.

Therefore, $p - q = 2 - 1 = 1$.

Correct Answer: 1

AI Suggestion: Option B

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the definitions of order and degree of a differential equation to correctly identify them in the given equation and then perform a simple subtraction.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concepts of order and degree of a differential equation, rather than just recalling facts or applying a specific procedure.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of definitions related to differential equations as covered in the textbook.

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