Class CBSE Class 12 Mathematics Differential Equations Q #812
KNOWLEDGE BASED
APPLY
1 Marks 2023 MCQ SINGLE
The integrating factor for solving the differential equation $x\frac{dy}{dx}-y=2x^{2}$ is:
(A) $e^{-y}$
(B) $e^{-x}$
(C) $x$
(D) $\frac{1}{x}$
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Correct Answer: D

AI Tutor Explanation

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Step-by-Step Solution

  1. First, rewrite the given differential equation in the standard form: $\frac{dy}{dx} + P(x)y = Q(x)$

    Given: $x\frac{dy}{dx} - y = 2x^2$

    Divide by $x$: $\frac{dy}{dx} - \frac{1}{x}y = 2x$

  2. Now, identify $P(x)$ and $Q(x)$.

    Here, $P(x) = -\frac{1}{x}$ and $Q(x) = 2x$

  3. Calculate the integrating factor (IF) using the formula: $IF = e^{\int P(x) dx}$

    $IF = e^{\int -\frac{1}{x} dx} = e^{-\int \frac{1}{x} dx} = e^{-\ln|x|} = e^{\ln|x^{-1}|} = \frac{1}{x}$

Correct Answer: 1/x

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 procedure of finding the integrating factor after converting the given differential equation into the standard form.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to find the integrating factor of a given differential equation. This involves algebraic manipulation and application of a formula.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of differential equations and the procedure to find the integrating factor, which is covered in the textbook.

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