Q646: $x~\log~x\frac{dy}{dx}+y=2~\log~x$ is an example of a :...

Class CBSE Class 12 Mathematics Differential Equations Q #646

KNOWLEDGE BASED

UNDERSTAND

1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE

$x~\log~x\frac{dy}{dx}+y=2~\log~x$ is an example of a :

(A) variable separable differential equation.

(B) homogeneous differential equation.

(D) differential equation whose degree is not defined.

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Correct Answer: C

AI Tutor Explanation

Step-by-Step Solution

The given differential equation is: $x \log x \frac{dy}{dx} + y = 2 \log x$

To determine the type of differential equation, we can rewrite it in the standard form for a first-order linear differential equation:

$\frac{dy}{dx} + P(x)y = Q(x)$

Divide the given equation by $x \log x$:

$\frac{dy}{dx} + \frac{1}{x \log x}y = \frac{2 \log x}{x \log x}$

$\frac{dy}{dx} + \frac{1}{x \log x}y = \frac{2}{x}$

Here, $P(x) = \frac{1}{x \log x}$ and $Q(x) = \frac{2}{x}$. Since the equation can be written in the form $\frac{dy}{dx} + P(x)y = Q(x)$, it is a first-order linear differential equation.

Correct Answer: first order linear differential equation.

AI Suggestion: Option C

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Pedagogical Audit

Bloom's Analysis: This is an UNDERSTAND question because the student needs to recognize the form of the given differential equation and classify it based on its properties. They need to understand the characteristics of different types of differential equations to identify the correct one.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the concept of different types of differential equations (variable separable, homogeneous, linear) and their standard forms. It's not just about recalling a definition but understanding the underlying concepts to classify the given equation.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of different types of differential equations as covered in the textbook and syllabus. It does not require application to a novel situation or case study.

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