Class CBSE Class 12 Mathematics Differential Equations Q #1716
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
The integrating factor of the differential equation $2x\frac{dy}{dx}-y=3$ is
(A) $\sqrt{x}$
(B) $\frac{1}{\sqrt{x}}$
(C) $e^{x}$
(D) $e^{-x}$
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Correct Answer: B

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Detailed Solution

Step 1: Standard Form Conversion

The given differential equation is 2x(dy/dx) - y = 3. To find the integrating factor, we must first write the equation in the standard linear form dy/dx + P(x)y = Q(x). Dividing the entire equation by 2x, we get: $$ \frac{dy}{dx} - \frac{1}{2x}y = \frac{3}{2x} $$

Step 2: Identify P(x)

Comparing the equation with the standard form, we identify P(x) = -1/(2x).

Step 3: Calculate Integrating Factor (IF)

The formula for the integrating factor is IF = e^∫P(x)dx. Substituting P(x): $$ IF = e^{\int -\frac{1}{2x} dx} = e^{-\frac{1}{2} \ln|x|} $$ Using logarithmic properties a ln(b) = ln(b^a): $$ IF = e^{\ln(x^{-1/2})} = x^{-1/2} = \frac{1}{\sqrt{x}} $$

Final Answer: (B)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must identify the standard form of a linear differential equation and apply the specific formula for the integrating factor.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the execution of a specific algorithmic process (standardization followed by integration) to reach the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to manipulate algebraic expressions within the framework of calculus, a core competency for the Differential Equations unit.

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